INVESTIGATION OF THREE-LEVELFINITE-DIFFERENCE TIME-DOMAINMETHODS FORMULTIDIMENSIONAL ACOUSTICSAND ELECTROMAGNETICSbyBrian Thao NguyenA dissertation submitted in partial ful�llmentof the requirements for the degree ofDoctor of Philosophy(Aerospace Engineering and Computational Science)in The University of Michigan1996Doctoral Committee:Professor P.L. Roe, ChairProfessor K. PowellProfessor B. van LeerProfessor J. Volakis

Dedicated with love and respect to my parents.

ii

ACKNOWLEDGEMENTSI would like to sincerely thank the following people and organizations for makingmy doctoral program truly excellent.My advisor, Phil Roe, who was truly a mentor. I have so many reasons to thankProfessor Roe that it would take at least another chapter to list.My committee members, Phil Roe, Bram van Leer, Ken Powell and John Volakisfor all their very helpful perspectives and critiques on my dissertation and at mydefense.Professors Van Leer, Powell and Roe, for supporting me towards the ultimateeducational experience as a research assistant my �rst year here. Their doors arealways open for my questions, and I am very grateful to have been able to count onthem.The Department of Energy and its Computational Science Graduate FellowshipProgram for their generous support for most of my doctoral program. This workwas supported in part by the Computational Science Graduate Fellowship Programof the O�ce of Scienti�c Computing in the Department of Energy.Department of Aerospace Engineering secretaries Bonnie Willey and MargaretFillion for being really friendly and taking interest in my personal life.The other hard working, stressed out CFD students for inspiring me to do mybest work. Because misery loves company, thanks to Sami Bayuk for company inthe lab on the 1994 New Year's Eve, Shawn Brown for sharing the dreadful monthsiii

before prelims and Je� Thomas for understanding instabilities.Professor Ken Powell for starting the University of Michigan Ballroom DanceClub. Rob Lowrie for taking me mountain biking (and for not getting mad at me fornearly breaking my neck). Fellow Head Banger and badminton jock Mohit Arora forthe exhausting workouts. Members of the CFD group for the stimulating discussions,technical and otherwise. Without these people, I would have no hope of being agenuinely multidimensional Doctor of Philosophy.I've always wanted to express my sincere gratitude to the three forces that shapedme as a person. At the top, being born and raised my parents certainly has changedthings for me. And their emphasis on education turned out to be a good idea afterall! Secondly, I thank the United States of America and its citizens for for givingme a new life when I was uprooted from my birthplace of Vietnam. I am grateful tohave had the opportunity to study and learn to think in America. Thirdly, thanksto Jo Dee Hunt, who taught me to appreciate the great fortunes in my life.My dearest and awesome friend Davee Kaplan who kept me stable and consistent,and therefore, convergent, not just through my doctoral program but through mymaster's degree also.Lynn Tuttle for making the time I did not spend working on my dissertation asrich and valuable as the time I did, for going above and beyond the call of duty toproofread my dissertation, for her faith in me and for seeing me the way I've alwayswanted to be seen.iv

TABLE OF CONTENTSDEDICATION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iiACKNOWLEDGEMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : iiiLIST OF FIGURES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : viiiLIST OF TABLES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xvLIST OF SYMBOLS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :xviiiLIST OF APPENDICES : : : : : : : : : : : : : : : : : : : : : : : : : : : : xxCHAPTERI. INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : : : 11.1 The Hierarchy of Techniques : : : : : : : : : : : : : : : : : : 11.2 Motivation for New Time-Domain Methods : : : : : : : : : : 41.3 Previous Work on Time-Domain Methods : : : : : : : : : : : 61.4 Overview of Dissertation : : : : : : : : : : : : : : : : : : : : : 11II. EQUATIONS, VARIABLES AND NOTATIONS : : : : : : : 132.1 Normalized Equations : : : : : : : : : : : : : : : : : : : : : : 132.2 A Note on the Notations : : : : : : : : : : : : : : : : : : : : 162.3 Duality for Two-Dimensional Problems : : : : : : : : : : : : 172.4 Conservative and Planar Wave Formulations : : : : : : : : : 192.5 Bicharacteristics Equations for Electromagnetics : : : : : : : 212.5.1 Re ection and Transmission Coe�cients : : : : : : 242.5.2 Wave Interaction at a Material Interface : : : : : : 262.6 Bicharacteristics Equations for Acoustics : : : : : : : : : : : : 282.6.1 Re ection and Transmission Coe�cients : : : : : : 292.6.2 Wave Interaction at a Material Interface : : : : : : 302.7 One-Dimensional Advection : : : : : : : : : : : : : : : : : : : 312.8 Normalized De�nitions of Numerical Errors : : : : : : : : : : 32III. SCHEMES IN ONE DIMENSION : : : : : : : : : : : : : : : : 37v

3.1 Reversible Central-Di�erence Schemes : : : : : : : : : : : : : 383.2 Reversible Upstream-biased Schemes : : : : : : : : : : : : : : 493.3 Reversibility : : : : : : : : : : : : : : : : : : : : : : : : : : : 593.4 Derivation of 1D Linear Schemes : : : : : : : : : : : : : : : : 613.5 Irreversible Central-Di�erence Schemes : : : : : : : : : : : : : 623.6 Irreversible Upstream-biased Schemes : : : : : : : : : : : : : 683.7 An Optimized Irreversible Upstream-biased Scheme : : : : : : 773.7.1 The Fourth-Order ul1dAd-O4 Scheme: w = 0 : : : : 793.7.2 Fifth-Order Scheme: w = �12 + 15� + �5 : : : : : : : : 823.7.3 Optimizing in w : : : : : : : : : : : : : : : : : : : : 853.8 The Next Step : : : : : : : : : : : : : : : : : : : : : : : : : : 89IV. SECOND-ORDER STANDARD AND UPWIND LEAPFROGSCHEMES IN TWO AND THREE DIMENSIONS : : : : : : 934.1 Two-Dimensional Acoustics and Electromagnetics : : : : : : : 934.1.1 Standard leapfrog : : : : : : : : : : : : : : : : : : : 944.1.2 Upwind leapfrog : : : : : : : : : : : : : : : : : : : : 954.1.3 Numerical Errors and Comparisons : : : : : : : : : 984.2 Three-Dimensional Acoustics : : : : : : : : : : : : : : : : : : 1014.2.1 Standard leapfrog : : : : : : : : : : : : : : : : : : : 1024.2.2 Upwind leapfrog : : : : : : : : : : : : : : : : : : : : 1024.2.3 Numerical Errors and Comparisons : : : : : : : : : 1044.3 Three-Dimensional Electromagnetics : : : : : : : : : : : : : : 1084.3.1 Standard leapfrog : : : : : : : : : : : : : : : : : : : 1084.3.2 Upwind leapfrog : : : : : : : : : : : : : : : : : : : : 1094.3.3 Numerical Errors and Comparisons : : : : : : : : : 1114.4 Other Considerations : : : : : : : : : : : : : : : : : : : : : : 114V. HIGH-ORDERLEAPFROG SCHEMES IN TWOAND THREEDIMENSIONS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1165.1 Three-Dimensional Acoustics : : : : : : : : : : : : : : : : : : 1175.1.1 Standard leapfrog : : : : : : : : : : : : : : : : : : : 1175.1.2 Upwind leapfrog : : : : : : : : : : : : : : : : : : : : 1225.2 Three-Dimensional Electromagnetics : : : : : : : : : : : : : : 1335.2.1 Standard leapfrog : : : : : : : : : : : : : : : : : : : 1335.2.2 Upwind leapfrog : : : : : : : : : : : : : : : : : : : : 1385.3 Two-Dimensional Acoustics and Electromagnetics : : : : : : : 149VI. EFFICIENCY : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 158vi

VII. NUMERICAL EXPERIMENTS : : : : : : : : : : : : : : : : : : 1727.1 One-Dimensional Advection : : : : : : : : : : : : : : : : : : : 1737.2 Two-Dimensional In�nite Radiating Current : : : : : : : : : : 1827.2.1 Global Distribution of Errors : : : : : : : : : : : : : 1847.2.2 Single-Point Histories : : : : : : : : : : : : : : : : : 1907.3 Three-Dimensional Pressure Source : : : : : : : : : : : : : : : 1917.4 Three-Dimensional Dipole Source : : : : : : : : : : : : : : : : 193VIII. FINAL WORDS : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2098.1 Summary and Conclusions : : : : : : : : : : : : : : : : : : : 2098.2 Further Work : : : : : : : : : : : : : : : : : : : : : : : : : : : 214APPENDICES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 217BIBLIOGRAPHY : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 237

vii

LIST OF FIGURESFigure2.1 Wave re ection and transmission at a non-conducting material in-terface. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 252.2 Wave interaction at grid cell interface. : : : : : : : : : : : : : : : : : 282.3 Total error vs. normalized error. : : : : : : : : : : : : : : : : : : : : 353.1 Standard leapfrog stencil for second-order accurate 1D advection,scheme sl1dAd-O2. : : : : : : : : : : : : : : : : : : : : : : : : : : : 403.2 Error plots for scheme sl1dAd-O2. : : : : : : : : : : : : : : : : : : : 423.3 Dispersion curve of standard leapfrog scheme for 1D advection. : : : 433.4 Reversible central-di�erence stencils for 1D advection. : : : : : : : : 463.5 Error plots for scheme sl1dAd-O4. : : : : : : : : : : : : : : : : : : : 473.6 Error plots for scheme sl1dAd-O6. : : : : : : : : : : : : : : : : : : : 483.7 Upwind leapfrog stencil for second-order accurate 1D advection, schemeul1dAd-O2. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 513.8 Error plots for scheme ul1dAd-O2. : : : : : : : : : : : : : : : : : : : 533.9 Reversible upstream-biased stencils for 1D advection. : : : : : : : : 553.10 Error plots for scheme ul1dAd-O4 : : : : : : : : : : : : : : : : : : : 563.11 Error plots for scheme ul1dAd-O6 : : : : : : : : : : : : : : : : : : : 573.12 Irreversible central-di�erence stencils for 1D advection. : : : : : : : 633.13 Error plots for scheme ciA. : : : : : : : : : : : : : : : : : : : : : : : 64viii

3.14 Error plots for scheme ciB. : : : : : : : : : : : : : : : : : : : : : : : 653.15 Irreversible upstream-biased stencils for 1D advection. : : : : : : : : 703.16 Error plots for scheme uiA. : : : : : : : : : : : : : : : : : : : : : : : 713.17 Error plots for scheme uiB. : : : : : : : : : : : : : : : : : : : : : : : 723.18 Error plots for scheme uiC. : : : : : : : : : : : : : : : : : : : : : : : 733.19 Error plots for scheme uiD. : : : : : : : : : : : : : : : : : : : : : : : 743.20 Stencil for wl schemes. : : : : : : : : : : : : : : : : : : : : : : : : : 803.21 Error plots for scheme wl4. : : : : : : : : : : : : : : : : : : : : : : : 813.22 w(�) for various optimizations of wl schemes. : : : : : : : : : : : : : 833.23 Error plots for scheme wl5. : : : : : : : : : : : : : : : : : : : : : : : 843.24 Excess ampli�cation of spurious mode of scheme wl5. : : : : : : : : 863.25 Amplitude and phase errors of wl scheme at N = 4. : : : : : : : : : 873.26 Region of small amplitude and phase errors for wl scheme at N = 4. 883.27 Error plots for scheme wlC. : : : : : : : : : : : : : : : : : : : : : : 903.28 Excess ampli�cation of spurious mode of scheme wlC. : : : : : : : : 914.1 Standard leapfrog stencil for 2D acoustics equations. : : : : : : : : : 954.2 Upwind leapfrog stencil for 2D bicharacteristic equations. : : : : : : 964.3 Phase error plots for scheme sl2dAc-O2. : : : : : : : : : : : : : : : : 994.4 Phase error plots for scheme ul2dAc-O2. : : : : : : : : : : : : : : : 994.5 Standard leapfrog storage scheme for 3D acoustics equations. : : : : 1034.6 Upwind leapfrog storage scheme for 3D acoustic bicharacteristic equa-tions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 104ix

4.7 De�nition of parametric angles in 3D. : : : : : : : : : : : : : : : : : 1054.8 Phase error plots for scheme sl3dAc-O2. : : : : : : : : : : : : : : : : 1064.9 Phase error plots for scheme ul3dAc-O2. : : : : : : : : : : : : : : : 1064.10 Standard leapfrog storage scheme for 3D electromagnetics equations. 1094.11 Upwind leapfrog storage scheme for 3D electromagnetic bicharacter-istic equations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1114.12 Phase error plots for scheme sl3dEm-O2. : : : : : : : : : : : : : : : 1124.13 Phase error plots for scheme ul3dEm-O2. : : : : : : : : : : : : : : : 1124.14 Di�erence in �p between schemes ul3dAc-O2 and ul3dEm-O2. : : : : 1145.1 Stencils for sl3dAc schemes. : : : : : : : : : : : : : : : : : : : : : : 1195.2 Phase error plots for scheme sl3dAc-O4. : : : : : : : : : : : : : : : : 1205.3 Phase error plots for scheme sl3dAc-O6. : : : : : : : : : : : : : : : : 1215.4 Stencils for ul3dAc schemes. : : : : : : : : : : : : : : : : : : : : : : 1285.5 Phase error plots for scheme ul3dAc-O4-A. : : : : : : : : : : : : : : 1295.6 Phase error plots for scheme ul3dAc-O4-B. : : : : : : : : : : : : : : 1295.7 Phase error plots for scheme ul3dAc-O4-C. : : : : : : : : : : : : : : 1305.8 Phase error plots for scheme ul3dAc-O4-D. : : : : : : : : : : : : : : 1305.9 Comparison of �p between ul3dAc-O4-C and ul3dAc-O4-D : : : : : 1335.10 Stencils for sl3dEm schemes. : : : : : : : : : : : : : : : : : : : : : : 1365.11 Phase error plots for scheme sl3dEm-O4. : : : : : : : : : : : : : : : 1375.12 Phase error plots for scheme sl3dEm-O6. : : : : : : : : : : : : : : : 1375.13 Stencils for ul3dEm schemes. : : : : : : : : : : : : : : : : : : : : : : 1415.14 Phase error plots for scheme ul3dEm-O4-A. : : : : : : : : : : : : : : 143x

5.15 Excess ampli�cation for scheme ul3dEm-O4-A. : : : : : : : : : : : : 1435.16 Phase error plots for scheme ul3dEm-O4-B. : : : : : : : : : : : : : : 1455.17 Excess ampli�cation for scheme ul3dEm-O4-B. : : : : : : : : : : : : 1455.18 Phase error plots for scheme ul3dEm-O4-C. : : : : : : : : : : : : : : 1465.19 Excess ampli�cation for scheme ul3dEm-O4-C. : : : : : : : : : : : : 1475.20 Phase error plots for scheme ul3dEm-O4-D. : : : : : : : : : : : : : : 1475.21 Excess ampli�cation for scheme ul3dEm-O4-D. : : : : : : : : : : : : 1495.22 Phase error plots for scheme sl2dAc-O4. : : : : : : : : : : : : : : : : 1515.23 Phase error plots for scheme sl2dAc-O6. : : : : : : : : : : : : : : : : 1515.24 Phase error plots for scheme ul2dAc-O4-A. : : : : : : : : : : : : : : 1525.25 Phase error plots for scheme ul2dAc-O4-B. : : : : : : : : : : : : : : 1525.26 Phase error plots for scheme ul2dAc-O4-C. : : : : : : : : : : : : : : 1535.27 Phase error plots for scheme ul2dAc-O4-D. : : : : : : : : : : : : : : 1535.28 Stencil for scheme ul2dAc-O4-B at � = 12 . : : : : : : : : : : : : : : : 1576.1 Computational resource requirements for 2D acoustics schemes. : : 1686.2 Computational resource requirements for 3D acoustics schemes. : : 1696.3 Computational resource requirements for 3D electromagnetics schemes.1707.1 1D pulse advection result using scheme sl1dAd-O2. : : : : : : : : : 1737.2 1D pulse advection result using scheme ul1dAd-O2. : : : : : : : : : 1747.3 1D pulse advection result using scheme sl1dAd-O4. : : : : : : : : : 1747.4 1D pulse advection result using scheme ul1dAd-O4. : : : : : : : : : 1747.5 1D pulse advection result using scheme sl1dAd-O6. : : : : : : : : : 174xi

7.6 1D pulse advection result using scheme ul1dAd-O6. : : : : : : : : : 1757.7 1D pulse advection result using scheme ciA. : : : : : : : : : : : : : 1757.8 1D pulse advection result using scheme ciB. : : : : : : : : : : : : : 1757.9 1D pulse advection result using scheme uiA. : : : : : : : : : : : : : 1757.10 1D pulse advection result using scheme uiB. : : : : : : : : : : : : : 1767.11 1D pulse advection result using scheme uiC. : : : : : : : : : : : : : 1767.12 1D pulse advection result using scheme uiD. : : : : : : : : : : : : : 1767.13 1D pulse advection result using scheme wl5. : : : : : : : : : : : : : 1767.14 1D pulse advection result using scheme wlC. : : : : : : : : : : : : : 1777.15 1D wave advection result using scheme sl1dAd-O2. : : : : : : : : : : 1787.16 1D wave advection result using scheme sl1dAd-O4. : : : : : : : : : : 1787.17 1D wave advection result using scheme sl1dAd-O6. : : : : : : : : : : 1797.18 1D wave advection result using scheme ciA. : : : : : : : : : : : : : : 1797.19 1D wave advection result using scheme ciB. : : : : : : : : : : : : : : 1797.20 1D wave advection result using scheme ul1dAd-O2. : : : : : : : : : 1797.21 1D wave advection result using scheme ul1dAd-O4. : : : : : : : : : 1807.22 1D wave advection result using scheme ul1dAd-O6. : : : : : : : : : 1807.23 1D wave advection result using scheme uiA. : : : : : : : : : : : : : 1807.24 1D wave advection result using scheme uiB. : : : : : : : : : : : : : 1807.25 1D wave advection result using scheme uiC. : : : : : : : : : : : : : 1817.26 1D wave advection result using scheme uiD. : : : : : : : : : : : : : 1817.27 1D wave advection result using scheme wl5. : : : : : : : : : : : : : : 181xii

7.28 1D wave advection result using scheme wlC. : : : : : : : : : : : : : 1817.29 1D wave advection result using various schemes at N = 4. : : : : : : 1827.30 Instantaneous solution of 2D radiating wire problem. : : : : : : : : 1857.31 Instantaneous error of 2D radiating wire problem. : : : : : : : : : : 1867.32 Maximum error of 2D radiating wire problem using scheme sl2dAc-O2.1877.33 Maximum error of 2D radiating wire problem using scheme sl2dAc-O4.1877.34 Maximum error of 2D radiating wire problem using scheme sl2dAc-O6.1887.35 Maximum error of 2D radiating wire problem using scheme ul2dAc-O2.1887.36 Maximum error of 2D radiating wire problem using scheme ul2dAc-O4-A. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1897.37 Maximum error of 2D radiating wire problem using scheme ul2dAc-O4-B. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1897.38 Histories for 2D pressure source using scheme sl2dAc-O2. : : : : : : 1927.39 Histories for 2D pressure source using scheme sl2dAc-O4. : : : : : : 1937.40 Histories for 2D pressure source using scheme sl2dAc-O6. : : : : : : 1947.41 Histories for 2D pressure source using scheme ul2dAc-O2. : : : : : : 1957.42 Histories for 2D pressure source using scheme ul2dAc-O4-A. : : : : 1967.43 Histories for 2D pressure source using scheme ul2dAc-O4-B. : : : : 1977.44 Histories for 3D pressure source using scheme sl3dAc-O2. : : : : : : 1987.45 Histories for 3D pressure source using scheme sl3dAc-O4. : : : : : : 1997.46 Histories for 3D pressure source using scheme sl3dAc-O6. : : : : : : 2007.47 Histories for 3D pressure source using scheme ul3dAc-O2. : : : : : : 2017.48 Histories for 3D pressure source using scheme ul3dAc-O4-A. : : : : 202xiii

7.49 Histories for 3D pressure source using scheme ul3dAc-O4-B. : : : : 2037.50 Histories for 3D dipole source using scheme sl3dEm-O2. : : : : : : : 2047.51 Histories for 3D dipole source using scheme sl3dEm-O4. : : : : : : : 2057.52 Histories for 3D dipole source using scheme sl3dEm-O6. : : : : : : : 2067.53 Histories for 3D dipole source using scheme ul3dEm-O2. : : : : : : : 2077.54 Histories for 3D dipole source using scheme ul3dEm-O2 at lower � = 13 .208B.1 Decoupling of the standard leapfrog scheme into Yee's scheme. : : : 225

xiv

LIST OF TABLESTable3.1 Error ranges for scheme sl1dAd-O2. : : : : : : : : : : : : : : : : : : 443.2 Error ranges for scheme sl1dAd-O4. : : : : : : : : : : : : : : : : : : 463.3 Error ranges for scheme sl1dAd-O6. : : : : : : : : : : : : : : : : : : 493.4 Error ranges for scheme ul1dAd-O2. : : : : : : : : : : : : : : : : : : 543.5 Error ranges for scheme ul1dAd-O4. : : : : : : : : : : : : : : : : : : 583.6 Error ranges for scheme ul1dAd-O6. : : : : : : : : : : : : : : : : : : 583.7 Error ranges for scheme ciA. : : : : : : : : : : : : : : : : : : : : : : 663.8 Error ranges for scheme ciB. : : : : : : : : : : : : : : : : : : : : : : 673.9 Error ranges for scheme uiA. : : : : : : : : : : : : : : : : : : : : : : 753.10 Error ranges for scheme uiB. : : : : : : : : : : : : : : : : : : : : : : 763.11 Error ranges for scheme uiC. : : : : : : : : : : : : : : : : : : : : : : 773.12 Error ranges for scheme uiD. : : : : : : : : : : : : : : : : : : : : : : 783.13 Error ranges for scheme uiE. : : : : : : : : : : : : : : : : : : : : : : 793.14 Error ranges for scheme wl4. : : : : : : : : : : : : : : : : : : : : : : 803.15 Error ranges for scheme wl5. : : : : : : : : : : : : : : : : : : : : : : 853.16 Error ranges for scheme wlC. : : : : : : : : : : : : : : : : : : : : : : 924.1 Error ranges for scheme sl2dAc-O2. : : : : : : : : : : : : : : : : : : 100xv

4.2 Error ranges for scheme ul2dAc-O2. : : : : : : : : : : : : : : : : : : 1004.3 Error ranges for scheme sl3dAc-O2. : : : : : : : : : : : : : : : : : : 1074.4 Error ranges for scheme ul3dAc-O2. : : : : : : : : : : : : : : : : : : 1074.5 Error ranges for scheme sl3dEm-O2. : : : : : : : : : : : : : : : : : : 1134.6 Error ranges for scheme ul3dEm-O2. : : : : : : : : : : : : : : : : : 1135.1 Error ranges for scheme sl3dAc-O4. : : : : : : : : : : : : : : : : : : 1225.2 Error ranges for scheme sl3dAc-O6. : : : : : : : : : : : : : : : : : : 1225.3 Error ranges for scheme ul3dAc-O4-A. : : : : : : : : : : : : : : : : 1315.4 Error ranges for scheme ul3dAc-O4-B. : : : : : : : : : : : : : : : : : 1325.5 Error ranges for scheme ul3dAc-O4-C. : : : : : : : : : : : : : : : : : 1325.6 Error ranges for scheme ul3dAc-O4-D. : : : : : : : : : : : : : : : : 1335.7 Error ranges for scheme sl3dEm-O4. : : : : : : : : : : : : : : : : : : 1385.8 Error ranges for scheme sl3dEm-O6. : : : : : : : : : : : : : : : : : : 1385.9 Error ranges for scheme ul3dEm-O4-A. : : : : : : : : : : : : : : : : 1445.10 Error ranges for scheme ul3dEm-O4-B. : : : : : : : : : : : : : : : : 1465.11 Error ranges for scheme ul3dEm-O4-C. : : : : : : : : : : : : : : : : 1485.12 Error ranges for scheme ul3dEm-O4-D. : : : : : : : : : : : : : : : : 1485.13 Comparison of 1D, 2D and 3D stability limits. : : : : : : : : : : : : 1505.14 Error ranges for scheme sl2dAc-O4. : : : : : : : : : : : : : : : : : : 1505.15 Error ranges for scheme sl2dAc-O6. : : : : : : : : : : : : : : : : : : 1545.16 Error ranges for scheme ul2dAc-A-O4. : : : : : : : : : : : : : : : : 1545.17 Error ranges for scheme ul2dAc-B-O4. : : : : : : : : : : : : : : : : : 155xvi

5.18 Error ranges for scheme ul2dAc-C-O4. : : : : : : : : : : : : : : : : : 1555.19 Error ranges for scheme ul2dAc-D-O4. : : : : : : : : : : : : : : : : 1566.1 Per cell computational requirement for 2D acoustics and electromag-netics schemes. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1616.2 Per cell computational requirement for 3D acoustics schemes. : : : : 1616.3 Per cell computational requirement for 3D electromagnetics schemes. 1626.4 E�ciency comparison for 2D acoustics schemes at �p = 1% : : : : : 1636.5 E�ciency comparison, 3D acoustics schemes at �p = 1% : : : : : : : 1636.6 E�ciency comparison, 3D electromagnetics schemes at �p = 1% : : 1646.7 E�ciency comparison for 2D acoustics schemes at �p = 1360 : : : : : 1656.8 E�ciency comparison, 3D acoustics schemes at �p = 1360 : : : : : : : 1666.9 E�ciency comparison, 3D electromagnetics schemes at �p = 1360 : : : 166

xvii

LIST OF SYMBOLSUnless otherwise stated, the symbols used in this dissertation have the followingde�nitions.Symbol De�nition Equation(s)A Jacobian matrix 2.33 2.63~B magnetic ux density 2.7c speed of propagation of sound or lightcex exact speed of propgation (c)cnum numerical speed of propagation� second-order accurate discrete central-di�erenceoperator~D electric ux density 2.5�� time step size�x; �y; �z grid spacings in each of three orthogonal directions~E electric �eld" electric permitivity�a normalized amplitude error 2.86�p normalized phase speed error 2.83F ux vector 2.30 2.31G G characteristic variable, scalar or vector 2.35 2.362.39 2.66~H magnetic �eldh grid spacing on a uniform gridI the imaginary number p�1Je normalized electric current 2.6Jm normalized magnetic current� wave number xviii

� coordinate along +bn (normal coordinate)` wavelength� eigenvalue (ampli�cation factor)� magnetic permeability� Courant number 2.81bn unit vector normal to a plane wave front (longitu-dinal direction) 2.27N resolution (`=h) 2.82P pressurep normalized pressure 2.11Q vector of dependent variables 2.23 2.25� non-dimensional wave number (�h) 2.91t transverse coordinate or plane� time coordinate or variable~V velocityV normalized velocity 2.12bx, by, bz unit vector in the x-, y- and z-directionZ intrinsic impedance (q�=" or �c)

xix

LIST OF APPENDICESAppendixA. EIGENSYSTEM OF THE ELECTROMAGNETIC JACOBIAN MA-TRIX : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 218A.1 Eigensystem Analysis : : : : : : : : : : : : : : : : : : : : : : 218A.2 Characteristic Variables : : : : : : : : : : : : : : : : : : : : : 220A.2.1 Alternate De�nitions : : : : : : : : : : : : : : : : : 222B. EQUIVALENCEOF YEE'S AND STANDARDLEAPFROG SCHEMES223C. METHOD OF ANALYSIS : : : : : : : : : : : : : : : : : : : : : : : : 229C.1 Classical von Neumann Analysis : : : : : : : : : : : : : : : : 229C.2 Numerical Computations of Eigensystems : : : : : : : : : : : 232D. DISCRETE FINITE DIFFERENCE OPERATORS : : : : : : : : : : 234

xx

CHAPTER IINTRODUCTIONPredicting the scattering of electromagnetic and acoustic waves has been the sub-ject of many ongoing basic and applied research e�orts. This thesis examines somenew approaches towards greater accuracy for the numerical time-domain simulationof electromagnetic and acoustic wave scattering.1.1 The Hierarchy of TechniquesVarious classical and numerical techniques have been developed for predictingelectromagnetic wave scattering.Some of the �rst techniques are ray tracing techniques, known as geometricaloptics (GO), which apply in the limits of very short wavelengths. Implicit in thename, these methods consider only specular re ections when a wave interacts with abody. They are limited by their disregard of di�raction, refraction and a multitude ofphysical e�ects. An extension to geometrical optics, known as physical optics (PO),introduces wave properties and surface currents such that spherical scattered wavesare generated when a wave strikes a surface. This spreads out the re ection so thatnot all the energy is focused in the specular direction. Physical optics still applyonly in the limits of small wavelengths. 1

2By themselves, GO and PO still poorly predict scattering by sharp edges. The-ories of di�raction are then incorporated into geometrical optics and physical opticsto improve the re ection when waves strike sharp edges. The geometrical theoryof di�raction (GTD) allows a ray to be di�racted equally in all directions when itstrikes an edge. The physical theory of di�raction (PTD) assumes a uniform currentsheet that is corrected for edge e�ects. However, GTD and PTD are not perfect�xes, and an additional �x, known as the equivalent current, is used to correct them.Regardless of the e�ectiveness of the �xes, methods based on optics are lim-ited by refraction and non-local phenomena. Methods for acoustics have analogouscounterparts to the above methods for electromagnetics, and they su�er the samedrawbacks. Enter frequency-domain methods into the hierarchy of techniques.The well-known method of moments and its analog in acoustics and uid dy-namics (panel method) assume localized sources, the sum of whose �elds and theimposed �eld (or free-stream ow) are required to satisfy the boundary conditionsof the problem. These requirements lead to a (usually large) set of linear equationsthat are solved for the source strengths. These are very useful methods in theirniches. Like the methods of ray tracing, however, they cannot account for non-localphenomena.In Harrington [1], methods were developed for the solutions of time-harmonicwaves by writing the �elds in terms of vector potentials that are governed by theHelmholtz equation. Exact solutions for varied but geometrically simple devices, suchas rectangular and circular wave guides and posts, are given as a superposition ofwave functions built up to satisfy the boundary conditions of the problem. Althoughthe solutions are exact, they often involve in�nite sums that are truncated in practice.Again, similar methods exist for acoustic scattering problems.

3When the problem geometry is too complex to solve analytically, numericaltechniques for partial di�erential equations can be employed to solve the resultingHelmholtz equations. Solving the actual partial di�erential equations means thatnon-local properties such as variations in the medium's parameters can be includedin the problem.Frequency-domain numerical techniques are abundant and have been used exten-sively. However, limitations imposed on their formulation have surfaced in increasingnumbers as more advanced analyses are required. One of two major approaches aretaken in frequency-domain methods. The method of moments [2] distributes sourcesover a body and requires that superposition of �elds from all sources and the exter-nal �eld satis�es a set of boundary conditions, usually taken at the sources. Fromthis, the source strengths are found. The numerical PDE methods discretizes theHelmholtz equations directly on a spatial mesh using �nite-element approximations.This generates a matrix problem which is then solved for the �elds. The entire classof frequency-domain methods essentially solves the Helmholtz equation instead ofMaxwell's curl equations or the acoustics divergence equations. As a result, when theconditions for the transformation to Helmholtz's equations do not apply, these meth-ods may not apply. Frequency-domain methods are not applicable in the presence ofnon-linearity. The presence of a non-linear device (boundary condition), even in lin-ear media, precludes their use. For broad band linear excitations, frequency-domainmethods must resolve all signi�cant frequencies individually, becoming more costlywith increasing bandwidth. Moreover, because these methods require signi�cantglobal operations they incur more communication overheads on massively parallelcomputers. Hence, their parallel e�ciency degrades faster than explicit time-domainalgorithms on massively parallel computers (see for example references [3, 4]).

4Time-domain methods have greater potential for solving complex problems thanfrequency-domain methods. Time-domain methods directly simulates the physicalsystem by making discrete approximations for the time and spatial derivatives to turnthe partial di�erential equations into a system of algebraic equations. Their simpli-fying assumptions are usually not an inherent requirement as are those assumptionsleading to Helmholtz's equation. Practically all assumptions are \optional" in time-domain methods, and most are in the constitutive relations. Complicated phenomenasuch as hysteresis and chiral behaviors can also be simulated simply by including thenumerical approximation of their e�ects into the time updates, though little workhas so far been done for these particular phenomena. (In chiral media, the equa-tion for electric density also contains the time derivative of the magnetic �eld, andthe equation for magnetic ux also contains the time derivative of the electric �eld(B.2).) For broad band scattering problems, it is preferable to use a time-domainmethod, since unlike frequency-domain methods, its cost does not grow with bandwidth.At the present time, time-domain methods lack the e�ciency of their frequency-domain counterparts (for problems where frequency-domain methods are applicable).Present time-domain computational methods for predicting electromagnetic scatter-ing are only hesitantly used on electrically large objects. Given a narrow-band, linearscattering problem, the time-domain methods generally result in signi�cantly longerrun time than a frequency-domain method.1.2 Motivation for New Time-Domain MethodsSince the 1980s, several general developments have motivated the on-going re-search in electromagnetic scattering analyses. The need to analyze radar cross-

5sections in the design of vehicles (such as military planes and missiles [5, 6, 7, 8]and ships) require electromagnetic simulation of the entire scatterer. Although thephysics involved in these problems are simple and can be modeled using linear, non-dispersive constitutive relationships, the problem sizes can be prohibitively large dueto the object size. Cavities such as inlets on these con�gurations require long in-tegration times (before initial transients subside) as well as �ne-grid resolutions (tocontrol the growth of errors during that time). For a �ghter design, problems reaching50 million grid points have resulted [9]. Such problems are not only computation-ally taxing, but unthinkable on any but the largest parallel computer platforms. Thecoming of age of the supercomputer is putting the solutions to these problems withinreach and have incited the search for e�cient computational methodologies.In contrast to the simple but computationally large problems, smaller but phys-ically complicated problems present a di�erent challenge. Complex physical phe-nomena arising from the interaction of electromagnetic waves and matter have madeincreasingly numerous appearances in the design of electronic, optical and electro-optical hardware. Optical switches, for example, require using non-linear material[10, 11]. Even chiral media have been suggested for use in waveguides [12].The recent recognition of ultra-wideband and impulse radar potential [13] bythe aerospace industry has also reiterated the need for more sophisticated analysis.The high peak power used in such systems leads to signi�cant non-linear behaviors.The ultra-wideband frequency content, ranging from d.c. to microwave, precludesany method using high- or low-frequency assumptions. The very broad band makesfrequency-domain solutions extremely infeasible, even if non-linear material behaviorcan somehow be removed (i.e. in free space).A less extreme and less exotic reason for a need to develop time-domain methods

6is for circuit compatibility analyses, where increasingly high speeds are used, gener-ating signi�cant cross-talk and thus precluding component-wise analyses. For theseproblems, frequency-domain methods may be impractical because of the non-linearresponses of the circuit components.To analyze the non-linear, anisotropic, chiral and dispersive characteristics of acon�guration, the electromagnetic wave phenomena must be simulated using time-domain predictive methods.Similar challenges exist in the �elds of acoustics and aeroacoustics. Great dis-crepancies of acoustic length scales, time scales and non-linear behaviors make theseproblems challenging. Yet they have important applications in aircraft noise control.Current research in time-domain scattering simulations are motivated by the needfor greater e�ciency to handle large problems and capability to simulate complicatedphysical interactions in the above examples. In numerical methods, e�ciency andaccuracy are often traded o� for one another. Other works regarding e�ciency dealwith e�cient implementations, particularly on massively parallel computers.1.3 Previous Work on Time-Domain MethodsThe �rst time-domain method for electromagnetics was introduced by Yee [14] in1966. This method computes electric and magnetic �elds that are staggered in spaceand time and can be interpreted as a standard leapfrog method. It is written for thelinear, non-dispersive media and Cartesian grid, is a very inexpensive method and isby far the most well-known method for electromagnetic scattering.Starting in the late 1980's, perhaps spurred by rapidly rising computational powerand advances made in related �elds (such as absorbing boundary conditions [15, 16,17]), a variety of numerical techniques have been o�ered. These can be categorized

7into several groups.Yee's scheme was proven to be second-order convergent on a nonuniform meshby Monk and Suli [18]. To address other issues of Yee's grid, much work using Yee'smethod (particularly that of Ta ove [5]) truncates the grid along grid lines thatapproximate physical boundaries, giving rise to a \stair-case" representation of theboundary where they do not coincide with grid lines. The works of Holland [19] in1983, Madsen [20] in 1988, Fusco [21] in 1990, Fusco et. al. [22] in 1991 and Yee [23]in 1992 were aimed strictly at generalizing Yee's original algorithm|mathematically,implementationally or both|for body-conforming grids. Holland o�ered the trans-formation of Maxwell's equation to curvilinear coordinates and the subsequent dis-cretization that is analogous to the original, except that it is on a curvilinear grid.This transformation is the same as that employed in computational uid dynamics(CFD) starting some 15 years earlier. Several applications of this technique werepresented by Fusco, but few others have presented results using Holland's scheme.Madsen o�ered several varieties of �nite-element and one �nite-volume method ondistorted grids. These methods primarily interpolate to obtain unknown quantitiesas required by the numerical scheme. Such interpolation techniques bring new errorsinto the scheme. The more general techniques are a great deal more costly thanthe original scheme, in terms of memory and computation time. Yee's solution tothe staircase e�ect was to employ the generalized grid method on a small body-conforming grid embedded in a global rectangular grid that uses the original scheme.The local and global grids are coupled by interpolation of the solution at each timestep. Again, the interpolation brings in new errors. However these error sources arelocalized to a small region of the computational domain. This method is not unlikethe Chimera grid embedding scheme [24, 25] used in CFD. It takes advantage of the

8great simplicity and low cost of the original scheme but incurs the overhead for thelogic in interpolation between the body-conforming grid and the global grid.It would be di�cult to compare the merits of the rectangular grid schemes withthose of the generalized grid schemes. Rectangular grid schemes are generally muchless expensive than their general grid extensions or counterparts. To compare, onemay have to weigh capability, complexity and cost (both computational and user) inaddition to accuracy. Any error due to the use of staircasing or collar grid interpo-lation may well be out-weighted by an argument of cost. In addition, at the presenttime, there is a consensus in the research community that second-order schemes haveinsu�cient accuracy and that high-order schemes are necessary. But as of yet, littleif any work has been published toward a high-order scheme on general grids. Forthese reasons, the decision of which to use is highly problem-dependent.Some new schemes have also started with Yee's scheme but were extended forgreater accuracy rather than for geometry. These included the fourth-order accurateextension of Fang [26] in 1989 and Deveze [27]. More recently, work on multiresolu-tion time-domain techniques have been published by Krumpholz [28, 29].Rather than starting with Yee's scheme, some completely new schemes were alsodeveloped. Two are the �nite-element scheme of Cangellaris et. al. [30] and the�nite-volume scheme Shankar [31, 32]. Both schemes are applicable on general grids.Shankar's scheme uses a Riemann solver to compute the ux on the grid cell interfacesand is the �rst scheme to use in electromagnetics the characteristic principles whichhave proved very popular in solving the Navier-Stokes equations. A third approach,the fractional-step method of Shang [33] is also a characteristic-based method. InCFD, the use of characteristic principles in deriving numerical schemes for hyper-bolic equations has been credited with sharply resolving discontinuities, reducing

9interference with physical di�usion, eliminating the need for questionable arti�cialdissipation, easier implementation of boundary conditions and last but not least,lower phase speed errors.An area of work from CFD unrelated to characteristic principles, but is oftenused along with characteristic principles, is ux limiting. In a landmark paper,Godunov [34] shows that any linear scheme with better than �rst-order accuracy willyield oscillatory solutions. These become highly visible and bothersome near highfrequencies and discontinuities such as shock waves. The �rst responses to Godunov'sdiscovery were given by Van Leer [35, 36], in which non-linear solution limiters areused to prevent new extrema from forming in the reconstruction. Solution limitingis now widely used in uid dynamics and can easily be applied to electromagneticand acoustic schemes.Although not expected in present practice, it is possible to generate an electro-magnetic shock wave [37] that behaves similarly to the uid-dynamic shock wave andcan bene�t greatly from characteristic based schemes and ux limiting.The ux-corrected transport (FCT) technique developed by Boris and Book [38]is another non-linear technique, but it is not designed for use with characteristicmethods. In FCT, a high-order oscillatory method and a low-order smooth methodare combined with a weight that favors the high-order method as much as possibleuntil new extrema would be generated in the next time step. The FCT technique didnot gain in CFD the popularity or following received by the ux-limiting technique.Nevertheless, it works well for unsteady problems such as time-domain scatteringsimulations. The FCT technique has been implemented in a CFD scheme by Zalesak[39]. In electromagnetics, FCT has been applied by Omick and Castillo [40, 41]to propagation of the step function wave form encountered in nuclear generated

10electromagnetic pulses.Although little work has been done toward modeling complicated physical inter-actions with material, dispersive media have been treated by Mohammadian et. al.[42] and Goorjian [43, 44]. These works use convolution integral extensions to exist-ing schemes. Mohammadian uses the scheme of Shankar, and Goorjian uses Yee'sscheme. The works of Goorjian have also treated non-linear dispersion.Work has also been progressing in time-domain acoustic scattering; however,little overlap in e�orts have occurred. There are some main di�erences betweenthe goals and challenges of acoustics and electromagnetics. Acoustic problems arefree from physical dispersion1 and anisotropy. Acoustic problems may have to dealwith media that move with a signi�cant fraction of the speed of propagation (speedof sound), or even faster. These problems are actually separated into their owncategory, called aeroacoustics, since the moving media bring on a tremendous amountof complications to say the least. Some work from acoustics proves to be useful inthe �eld of aeroacoustics, but this concern is absent in the electromagnetics �eld.The commonality between electromagnetics and acoustics is the mathematicalsimilarities between the sets of governing equations. Dispersive and di�usive er-rors are concerns to both. In numerical schemes, both would strive to resolve highfrequency waves as e�ciently as possible.There are many more new algorithmic developments in acoustics, but no schemeor class of schemes has enjoyed a following on the scale of Yee's scheme in elec-tromagnetics. This may be in part because the greatest challenge is presently notsimple acoustics but aeroacoustics. This involves multi-dimensional linear advection,1all acoustic waves move at the speed of sound regardless of frequency, and only shock wavesmove faster|but shock waves are not considered an acoustic problem for a number of complexitiesinvolved in them

11a problem for which Yee's scheme shows no remarkable potential compared to themultitude of schemes in CFD.The aeroacoustics problem is currently being attacked by many e�orts. Tam andWebb [45] formulated the dispersion-relation-preserving (DRP) schemes in whichoptimization was done to reduce an integral measure of the error in the numericalphase angle and the exact phase angle. High resolution schemes, including ENO(essentially non-oscillatory), schemes are also being brought into use.While work was carried out in search of more accurate techniques, many neglectedto include the cost of using more advanced schemes. Any e�ort to put these schemesto practical use would want to ask whether the same accuracy can be attained forthe same cost just by using the simpler schemes with denser grids. One review ofcost e�ciency was the article by Shlager [46] for several schemes. The quantitativeresults of cost e�ciency analysis may not be the only thing in uencing the choiceof schemes used in a certain code. Other factors that may be considered are ease ofuse (which in a sense, a�ects the human resource cost), parallelizability (wheneverparallel implementations are considered), boundary condition issues and exibility.Various combinations of these considerations may a�ect the decision on the bestscheme to use.1.4 Overview of DissertationThe scope of this work is limited to the linear Maxwell's and acoustics equations.No aeroacoustic problem is treated but concurrent work on aeroacoustic using sim-ilar schemes can be found in reference [47]. This work is basic and mathematicalin nature, focusing on the exploration of some potentially useful rectangular gridmethods in electromagnetics and acoustic scattering and comparing them with each

12other. In particular, it examines at a new and an old type of di�usion-free schemes.The old type is Yee's scheme, including new high-order extensions. The new typeis the characteristics-based \upwind leapfrog" scheme, including high-order exten-sions. The goal of this work ideally is to �nd e�cient discretization schemes for useon very large simulations. In the process, the two types of schemes are comparedand contrasted in detail.Chapter II of this dissertation describes the essential electromagnetics and acous-tics problems covered (including the discussion of the bicharacteristic equations), in-troduces the notations, describes the de�nitions used in this dissertation and stateshow the numerical errors are to be compared. Chapter III examines the fundamental1D schemes that are the building blocks of multi-dimensional schemes. Chapter IVpresents the new upwind leapfrog scheme applied to the electromagnetic and acous-tic problems in multi-dimensions. It analyzes the multi-dimensional second-orderaccurate standard leapfrog and upwind leapfrog schemes. These multi-dimensionalschemes are the bases of more complex schemes to come. Chapter V investigatesthe high-order accurate multi-dimensional schemes and how they perform for thesame problems. These are extensions to the second-order accurate schemes and areaimed at improving accuracy. We reserve discussions on the bottom-line e�ciencyof these schemes for Chapter VI where all the multi-dimensional schemes are consid-ered together. Chapter VII presents the results of some numerical experiments usingthe schemes of the previous chapters. The work is summarized and �nal words areo�ered in Chapter VIII.

CHAPTER IIEQUATIONS, VARIABLES AND NOTATIONS2.1 Normalized EquationsWe will �rst facilitate the treatment of two physical systems (electromagnetic andacoustic) by introducing the \normalized" variables. These are dependent variableswith the same dimensional units, allowing us to easily nondimensionalize them with asingle quantity, making the notations in the eigensystem and characteristic analysesless cluttered. We will also be able to speak of the electromagnetic or acousticequations interchangeably. The governing equations for normalized variables arecalled normalized equations. These transformations do not employ any physicalassumptions other than those which limit the scope of this work.The general form of Maxwell's equations relating the electric to the magnetic�elds are @ ~D@� �r� ~H+ ~J e = 0 (2.1)@ ~B@� +r� ~E = 0 (2.2)where � is the time coordinate, ~D is the electric ux density, ~B is the magnetic ux density, ~E is the electric �eld, ~H is the magnetic �eld and ~J e is the electriccurrent density. For linear, isotropic, non-chiral media1, which is the scope of this1In time-domain numerical techniques, it is possible to treat non-linearly dispersive media by13

14dissertation, they are supplemented by the constitutive relationships~D = "~E (2.3)~B = � ~H (2.4)where the permitivity " and the permeability � are scalar parameters of the medium.We de�ne the normalized variables E � ~E (2.5)Je � 1" ~J e (2.6)H � Z ~H (2.7)where the intrinsic admittance Y � q"=� � 1=Z, and Z is the intrinsic impedance.These new variables are substituted into Maxwell's equations to write the normalizedMaxwell's equations,@@� 0BB@ EH 1CCA+ c0BB@ �r�Hr�E 1CCA + 0BB@ JeJm 1CCA+ 0BB@ "�" Ec�c H 1CCA+ 0BB@ crZZ �H0 1CCA = 0 (2.8)where c = 1=p"� is the local speed of light. Note that the �ctitious normalizedmagnetic current Jm has been introduced to emphasize the symmetry between the�eld vectors and their governing equations and to give provision for a possible inputof magnetic current sources. Note that space vectors are denoted by an overheadarrow or by bold type and the subscript � denotes a derivative with respect to time.building upon techniques for linear media [44]. Anisotropic media are not within the scope of thiswork and extensions of schemes examined herein for them may or may not be possible. Numericalmethods that are developed for anisotropic media (particularly if the principal axes of anisotropydo not coincide with the coordinate axes) in general will be much more costly. Inhomogeneousmedia are modeled as locally homogeneous media so that locally, the source terms proportional tothe derivatives of the constitutive parameters or the speed of propagation are small and neglected.

15The somewhat similar acoustics equations are@P@� + �c2r � ~V = 0 (2.9)�@~V@� +rP = 0 (2.10)where P is the pressure disturbance, ~V = Vxbx + Vyby + Vzbz is the velocity vectordisturbance, � is the density if the medium and c is the speed of sound. Notethat quantities with analogous meanings for Maxwell's equations and the acousticsequations are represented by the same symbol. The speed of propagation c is such aquantity. Others are the intrinsic admittance and the intrinsic impedance Z (whichis Z � �c for acoustics). The use of these symbols emphasize the similarities betweenMaxwell's equations and the acoustics equations and should not cause any confusion.We de�ne for linear isotropic acoustic media, the normalized variablesp � P (2.11)V � ubx+ vby+ zbz = Z(Vxbx+ Vyby+ Vzbz) = Z~V (2.12)to write @@� 0BB@ pV 1CCA+ c0BB@ r �Vrp 1CCA+ 0BB@ �S 1CCA+ 0BB@ �crZZ �VZ�Z V 1CCA = 0 (2.13)We have again introduced �ctitious source terms � for the pressure and S for thevelocity to show duality with the electromagnetic system of equations (in 2D, seebelow) and to give a place for the input of source terms when they are used.The normalized variables (2.5, 2.7, 2.11, 2.12) and equations (2.8, 2.13) havesome advantages for this dissertation. They reveal the propagation speed c explicitly.This parameter is important in numerical analysis and in the derivation and use ofupstream-biased methods in this work. The normalized notations also reduce the

16clutter of coe�cients required to write equations with consistent units and are usefulwhen characteristic variables are introduced and used throughout this dissertation.2.2 A Note on the NotationsWe will be making extensive use of subscripts and superscripts in this disser-tation. Unless otherwise noted, they follow these conventions. Variables that arefunctions of space-time coordinates use upright superscripts as a part of their nameand slanted subscripts to refer to derivatives with respect to the coordinates. For ex-ample, G+xxxxyy = @G+x@x3@y2 is the third-order derivative in x and second-order derivativein y of the characteristic variable associated with the +x direction. (Characteris-tic variables are introduced below.) Variables that are functions of the grid indicesuse upright superscripts as a part of their name and slanted subscripts to refer tothe indices. For example, G+xn;i;j;k is the discrete variable G+x at the grid coordi-nates (n; i; j; k), and Exn;i;j;k is the discrete x-component of E at the grid coordinates(n; i; j; k). Constants and parameters (i.e., �, Z, c, etc.) that are (generally) notfunctions of any coordinate system use upright subscripts and superscripts as a partof their identity. For example, ZL and ZR refer to the intrinsic impedance of materialsto the left and right sides of a reference point.As mentioned above, spatial vectors used in this dissertation are denoted byeither bold-faced type or by an overhead arrow. For example, the following vectorsare written explicitly in terms of their components: E = (Ex; Ey; Ez) = (Ex; Ey; Ez),H = (Hx;Hy;Hz) = Z(Hx;Hy;Hz), and so forth.A phase-space vector represent point in phase space rather than a direction inthe spatial coordinates. They are designated with an underline. The variable Q inthis dissertation is such a vector, as are eigenvectors. The \scalar ux vector" is F

17(2.22), whereas the \vector ux vector" is ~F (2.29). Matrices are represented by adouble underline, such as the Jacobian matrix A.2.3 Duality for Two-Dimensional ProblemsThe divergence operator in the acoustics equations and the curl operator in theelectromagnetic equations are very di�erent in their full form, but in two dimensions,they are very much alike. In fact, the 2D acoustic equations with constant constituentparameters can be directly transformed into the 2D transverse magnetic (TM) or 2Dtransverse electric (TE) equations. In 2D equation (2.13) reduces top� + c �ux + vy�+� = 0 (2.14a)u� + cpx + Sx = 0 (2.14b)v� + cpy + Sy = 0 (2.14c)Using the transformation p$ Ez;�$ J e;z (2.15a)u$ �Hy; Sx $ �Jm;y (2.15b)v $ Hx; Sy $ Jm;x (2.15c)gives the 2D TM equationsEz� + c ��Hyx +Hxy �+ J e;z = 0 (2.16a)Hy� � cEzx + Jm;y = 0 (2.16b)Hx� + cEzy + Jm;x = 0 (2.16c)and using the transformation p $ Hz;�$ Jm;z (2.17a)

18u$ Ey; Sx $ J e;y (2.17b)v $�Ex; Sy $�J e;x (2.17c)gives the 2D TE equations Hz� + c �Eyx � Exy�+ Jm;z = 0 (2.18a)Ey� + cHzx + J e;y = 0 (2.18b)Ex� � cHzy + J e;x = 0 (2.18c)The duality exists between the acoustic system and the electromagnetic system onlyfor constant constituent parameters, and this is lost to some degree for variableconstituent parameters2. In two-dimensions, only the treatment of the acoustics2The 2D equations with variable constituent parameters arep� + c �ux + vy�+� � cZxu+ ZyvZ = 0 (2.19a)u� + cpx + Sx + Z�Z u = 0 (2.19b)v� + cpy + Sy + Z�Z v = 0 (2.19c)Ez� + c ��Hyx +Hxy�+ Je;z + "�" Ez + cZxHy � ZyHxZ = 0 (2.20a)Hy� � cEzx + Jm;y + c�c Hy = 0 (2.20b)Hx� + cEzy + Jm;x + c�c Hx = 0 (2.20c)and Hz� + c �Eyx � Exy�+ c�c Hz + Jm;z = 0 (2.21a)Ey� + cHzx + Je;y + "�" Ey + cZzHx � ZxHzZ = 0 (2.21b)Ex� � cHzy + Je;x + "�" Ex + cZyHz � ZzHyZ = 0 (2.21c)which do not share the duality properties of their constant constituent parameter forms. Onereason is that besides c, the acoustic equations have just one independent parameter (Z ) andthe electromagnetic equations have two independent parameters (Z and "). If the constituentparameters vary in space but not in time, the acoustic equations are dual to the TM equations, butnot to the TE equations.

19equations is presented in this dissertation. All schemes and results are identical forthe TM and TE equations.2.4 Conservative and Planar Wave FormulationsTo facilitate the simultaneous treatment of the electromagnetic and the acousticproblems and to show the characteristic analysis of their systems of equations, theequations (2.8) and (2.13) are cast in the same general conservative form@Q@� +r � ~F = S (2.22)where for electromagnetics, Q = 0BBBBBBBBBBBBBBBBBBBB@ ExEyEzHxHyHz1CCCCCCCCCCCCCCCCCCCCA (2.23)

~F = c0BBBBBBBBBBBBBBBBBBBB@0Hz�Hy0�EzEy

1CCCCCCCCCCCCCCCCCCCCA bx+ c0BBBBBBBBBBBBBBBBBBBB@ �Hz0HxEz0�Ex1CCCCCCCCCCCCCCCCCCCCA by + c0BBBBBBBBBBBBBBBBBBBB@

Hy�Hx0�EyEx01CCCCCCCCCCCCCCCCCCCCA bz (2.24)

20and for acoustics, Q = 0BBBBBBBBBBB@ puvw 1CCCCCCCCCCCA (2.25)~F = c0BBBBBBBBBBB@ up00 1CCCCCCCCCCCA bx+ c0BBBBBBBBBBB@ v0p0 1CCCCCCCCCCCA by+ c0BBBBBBBBBBB@ w00p 1CCCCCCCCCCCA bz (2.26)For a plane wave moving along any unit vectorbn = nxbx+ nyby+ nzbz (2.27)all spatial gradients are parallel to that unit vector (which is normal to the planarwave front). The r operator can be reduced to the longitudinal operatorrn = bn @@� (2.28)where � is the normal coordinate increasing along the direction +bn. Equation (2.22)then becomes @Q@� + @F@� = S (2.29)For Maxwell's equations, F = c0BBBBBBBBBBBBBBBBBBBB@ Hynz �HznyHznx �HxnzHxny �HynxEzny � EynzExnz � EznxEynx � Exny1CCCCCCCCCCCCCCCCCCCCA (2.30)

21and for the acoustics equations, F = c0BBBBBBBBBBB@ UPnxPnyPnz 1CCCCCCCCCCCA (2.31)where U = bn �V (2.32)2.5 Bicharacteristics Equations for ElectromagneticsThe characteristics analysis for Maxwell's equations begins with the Jacobianmatrix for the electromagnetic ux vector (2.30),A � @F@Q = c0BBBBBBBBBBBBBBBBBBBB@ 0 0 0 0 nz �ny0 0 0 �nz 0 nx0 0 0 ny �nx 00 �nz ny 0 0 0nz 0 �nx 0 0 0�ny nx 0 0 0 01CCCCCCCCCCCCCCCCCCCCA (2.33)which has the eigenvalues � = 0; 0; c; c;�c;�c (2.34)The eigensystem of this matrix is derived and discussed in Appendix A. Uponpremultiplying equation (2.29) by either of the left eigenvectors associated with � =0, lH0 (A.2) and by lE0 (A.3), and introducing the wave strengthsGE0 = lE0 �Q = bn �E (2.35)

22and GH0 = lH0 �Q = bn �H (2.36)we have @@� GE0 = �bn � Je (2.37)@@� GH0 = �bn � Jm (2.38)This states that any locally one-dimensional wave must be transverse since compo-nents of H and E along bn are steady (except for the �eld arising from the currents).Since the eigenvalues are repeated, any linear combination of the associated eigen-vectors is also an associated eigenvector. The choices (A.2, A.3) led to the separationof the electrical and magnetic �eld components along bn.A similar procedure for the moving modes leads to the characteristic variable (seeAppendix A) G+n = Et � bn�H2 (2.39)where the superscript +n denotes that G+n is associated with the positive longitu-dinal direction and is the variable moving in that direction. (Reversing the sign onbn gives the variable for the reverse direction G�n.) The superscript \t" denotes thecomponents transverse to bn. By de�nition, G�n are also vectors transverse to bn andhave the same polarization as E. The equation governing the characteristic combina-tion G+n is derived by combining the governing equations (or its constituent terms)in the same proportion. This procedure for the electromagnetic governing equations(2.8) gives the electromagnetic bicharacteristic equations" @@� + c @@�#G+n + Je;t � bn � Jm2 = crt(bn �E)� bn �r(bn �H)2 (2.40)The bicharacteristic governing equations have the advantage of presenting a system ofadvection equations with de�nite senses, speeds and directions of propagation, which

23clearly identi�es how the equations can be discretized in an upstream-biased manner(using an appropriate upstream-biased scheme such as one of the upwind leapfrogschemes to be introduced later). An in�nite number of characteristic variables andbicharacteristic equations can be found at a single point in space by varying bn con-tinuously. This means that it is possible to de�ne arbitrarily many characteristicvariables, each having an appropriate bicharacteristic governing equation. However,an over-constrained system of equations does not occur since each applies in a dif-ferent plane [48].Without the source terms and the cross terms, the bicharacteristic equations areexactly the 1D advection equations. The primitive variables E and H follow fromequation (2.39) by Et =G+n +G�n (2.41a)Ht = bn� (G+n �G�n) (2.41b)which can easily be computed in a numerical scheme.3Any simple wave can be represented by (2.39), with the wave along �bn taken tohave zero strength, (G�n = 0). The primitive variables represented by such a wave3Rather than using equation (2.41) to �nd the primitive solutions, it is common in CFD literatureto write a sum of eigenvectors, i.e., Q =X(l �Q)r (2.42)where the right and left eigenvectors are orthonormal. This is equivalent to equation (2.41) as thecomponents of G+n are formed from the inner product of l and Q. The summation is particularlyuseful because the summation can select terms from di�erent directions along bn, as consistent withupwinding. However, expressing Q as products of right and left eigenvectors can be somewhatawkward in the present case. As mentioned in Appendix A it is di�cult to �nd a general form ofthe eigenvectors that will work in all instances. First, it is impossible to choose four independenteigenvectors (for the four movingmodes) whose expressions are symmetric with respect to the threecoordinate axes. Second, after choosing one set of (right or left) eigenvectors, the inversion of theeigenvector matrix introduces terms involving one or more of 1nx , 1ny and 1nz , which is indeterminatefor arbitrary bn. Equation (2.41) can just as easily be used in upwinding when G+n and G�n arechosen appropriately. This is shown below in Section 2.5.2.

24are Et = G+n (2.43a)Ht = bn�G+n (2.43b)2.5.1 Re ection and Transmission Coe�cientsWhen a wave strikes the interface between two non-conducting media, part ofit is transmitted and the remainder is re ected back. Using the concept of thecharacteristic variable vector, the coe�cients of transmission and re ection can beobtained using concise notations.Figure 2.1 depicts the general wave striking the interface between material 1(top) and material 2 (bottom). The incident wave (denoted by \i"), re ected wave(denoted by \r") and transmitted wave (denoted by \t") are decomposed to theirtransverse-magnetic (TM) and their transverse-electric (TE) components given byGi = GTM;ibz+ GTE;ibni � bz (2.44a)Gr = GTM;rbz+ GTE;rbnr � bz (2.44b)Gt = GTM;tbz+ GTE;tbnt � bz (2.44c)Each of these represents a simple wave|there is no reverse wave. Assuming that the�elds in material 1 are represented by the sum of the incident and re ected wavesand the �elds in material 2 are represented by the transmitted wave, then each wavecontributes ~E = G (2.45)~H = Ybn �G (2.46)to the ~E and ~H �elds in proportion to its magnitude via (2.43). Summing thesefor each side of the interface then equating the components to satisfy the boundary

251 z x2 bz bz bnr � bzbni bnrTransmitted wavebz bntbnt � bzbni � bzIncident wave Re ected waveyGi or Gi Gr or Gr

Gt or GtFigure 2.1: Wave re ection and transmission at a non-conducting material interface.condition that components parallel to the interface must be equal, one obtains thevector equationsGTM;ibz+GTE;ini;ybx+GTM;rbz+GTE;rnr;ybx = GTM;tbz+GTE;tnt;ybx (2.47)Y1 �GTM;ini;ybx�GTE;ibz+GTM;rnr;ybx�GTE;rbz� = Y2 �GTM;tnt;ybx�GTE;tbz� (2.48)Without loss of generality, each wave can be assumed to have the functional formG(x; y; z; t) = f �xnx + yny + znzc � �� (2.49)Then the usual functional dependence argument can be made on the surface [1],resulting in the deduction ni;x = nr;x (2.50)and Snell's law, ni;xc1 = nt;xc2 (2.51)As usual, these relationships (and the fact that all bn are unit vectors) can be used touniquely determine bnr and bnt. Knowing the longitudinal directions bn for each wave,

26equation (2.47) and (2.48) can be solved, �nding the re ection and transmissioncoe�cients (wave strength ratios)GTM;rGTM;i = ni;yY1 � nt;yY2ni;yY1 + nt;yY2 (2.52)GTM;tGTM;i = 2Y1ni;yni;yY1 + nt;yY2 (2.53)GTE;rGTE;i = �nt;yY1 � ni;yY2nt;yY1 + ni;yY2 (2.54)GTE;tGTE;i = 2Y1ni;ynt;yY1 + ni;yY2 (2.55)Note that the sign convention chosen in (2.44a, 2.44b, 2.44c) led to the negative signin GTE;rGTE;i , which is opposite to that shown in many text books that use the reverseconvention.2.5.2 Wave Interaction at a Material InterfaceWe now show how waves interact at a non-conducting material interface. Thisexact analysis leads to how the bicharacteristic equations and variables are used ina numerical scheme.In the upwind leapfrog scheme for electromagnetics, the tangential componentsof H and E are de�ned on the faces in the grid. To advance the solution on a face,we use only the bicharacteristic equations associated with the face-normal direction,i.e. the wave direction bn is along the face normal, from the left to the right side. Oneach side, the equation for information moving toward the face is used, as shown inFigure 2.2. The waves bring information to the face from either sides. Although theactual waves may not be traveling normal to the face, right next to the interface,only the behavior of G�n needs to be studied. If the actual wave is not travelingalong bn, that is accounted for by the cross terms in the equations governing G�n.

27Directly from equations (2.35) and (2.36), we know�(E � bn) = �(H � bn) = 0 (2.56)across the interface, as expected of a transverse wave. We need no longer be con-cerned with these longitudinal components. Knowing the current solutions on theright and left sides of the face, we compute the two tangential components ofG�nR = EtR + bn �HR2 (2.57)and the two tangential components ofG+nL = EtL � bn�HL2 (2.58)using any method such as the numerical methods to be discussed later in this disser-tation. These are the pieces of information brought from the right and left sides ofthe interface (denoted by \R" and \L") as shown in Figure 2.2. The quantities withwhich they are associated are instantaneous quantities right next to the interface buton one side or the other.With these known wave strengths, we rewrite them in terms of the standardprimitive variables, G�n = ~Et � bn� Z ~H2 (2.59)and use the boundary condition ~EtR = ~EtL = ~Et (2.60a)~HtR = ~HtL = ~Ht (2.60b)at the non-conducting interface. The solutions satisfying these boundary conditionsare ~Et = 2ZR + ZL �ZRG+nL + ZLG�nR � (2.61a)

28G+nL G�nROutgoing waves G�nL R G+nRLIncoming wavesFigure 2.2: Wave interaction at grid cell interface. Incoming waves G�nR and G+nL arecomputed by the numerical scheme. After interacting, outgoing wavesG�nR and G+nL result.~Ht = 2ZR + ZL bn� �G+nL �G�nR � (2.61b)which lead to the outgoing wave strengthsG+nR = 1ZR + ZL �2ZRG+nL � (ZR � ZL)G�nR � (2.62a)G�nL = 1ZR + ZL �2ZLG�nR + (ZR � ZL)G+nL � (2.62b)to be used in the next iteration. Note that if the medium on both sides of theinterface has the same intrinsic impedance, the interface mathematically disappears,leaving the waves undisturbed as they pass through.2.6 Bicharacteristics Equations for AcousticsThe Jacobian matrix for the acoustic ux vector (2.31) isA � @F@Q = c0BBBBBBBBBBB@ 0 nx ny nznx 0 0 0ny 0 0 0nz 0 0 0 1CCCCCCCCCCCA (2.63)which has the eigenvalues �1;2;3;4 = 0; 0;�c (2.64)The stationary (shear) mode Riemann variable isG0 = bn�V (2.65)

29which is the transverse velocity vector. Unlike the electromagnetic system, the sta-tionary mode is a transverse vector and the moving modes are scalars. The travelingmode characteristic variable isG+n = p + bn �V2 = p+ U2 (2.66)governed by the equation" @@� + c @@�#G+n + �+ bn � S2 = �crt �V2 (2.67)which is derived in the same manner explained for electromagnetics bicharacteristicsgoverning equation (2.40). The primitive variables are the normalized pressure andthe longitudinal component of velocity,p = G+n +G�n (2.68a)U = G+n �G�n (2.68b)As in the electromagnetic system, any simple wave can be represented by (2.68), withthe wave along �bn taken to have zero strength. The primitive variables representedby such a wave are P = U = G+n (2.69)2.6.1 Re ection and Transmission Coe�cientsAlthough acoustics problems do not usually have discontinuous media, the piece-wise constant material assumption used when the media is inhomogeneous generatesdiscontinuities at the discrete cell interfaces. For this reason, and for completeness,the re ection and transmission coe�cients using the acoustic characteristic variablesare presented.

30An incident wave impinges on a material interface, giving rise to a re ected waveand a transmitted wave, as described in Figure 2.1. This time, the wave strengths arescalar variables and we will not be using the vectors bz and bn�bz. The same procedurefor the electromagnetic system is applied here, with the boundary condition beingthat pressure and normal velocity must be continuous across the interface. We get,for the acoustic system, the re ection coe�cientGrGi = ni;yY1 � nt;yY2ni;yY1 + nt;yY2 (2.70)and the transmission coe�cientGtGi = 2Y1ni;yni;yY1 + nt;yY2 (2.71)which are identical to those of the 2D TM system. This is not surprising, sincethe boundary conditions are the same as the 2D TM waves, upon applying thetransformation of variables (2.15).2.6.2 Wave Interaction at a Material InterfaceModeling a continuously varying media as a locally homogeneous media withmaterial discontinuity at the grid interfaces requires the solution to the problem ofthe G�n waves hitting the surface, as depicted in Figure 2.2. This is very similar tothe electromagnetic case in Section 2.5.2. Let the grid interface be normal to bn. Werewrite the acoustic characteristic variable asG�n = P � ZU2 (2.72)with U = bn � ~V (2.73)

31Consistent with their propagation directions, G+n is taken from the right side andG�n from the left side. Using the interface boundary conditionsUR = UL = U (2.74a)PR = PL = P (2.74b)and equation (2.72), we get the interface solutionsU = 2ZR + ZL �G+nL �G�nR � (2.75)P = 2ZR + ZL �ZRG+nL + ZLG�nR � (2.76)and the outgoing wave strengths areG+nR = 2ZR + ZL �2ZRG+nL � (ZR � ZL)G�nR � (2.77)G�nL = 2ZR + ZL �2ZLG�nR + (ZR � ZL)G+nL � (2.78)Note again that if ZR = ZL, the material interface mathematically disappears, andthe waves pass through it undisturbed.2.7 One-Dimensional AdvectionIn 1D, the bicharacteristic equations (2.40) and (2.67) directly reduce to the twoindependent advection equations, Gt + cGx = 0 (2.79a)St � cSx = 0 (2.79b)regardless of whether the original system of equations was acoustic or electromag-netic. Equations (2.79) independently govern two waves moving in the +x and �xdirections and has the solutionsG(x; t) = f �xc + t� (2.80a)

32S(x; t) = f �xc � t� (2.80b)where f is an arbitrary function.The 1D acoustic and electromagnetic equations can be transformed into the equa-tion set (2.79). Their behavior, whether exact or discretized in a linear scheme isidentical to the 1D advection equation. Therefore, in 1D, we examine only theadvection equation. The results are identical for the acoustic and electromagneticequations in either their standard forms (2.8) and (2.13) or their bicharacteristicforms (2.40) and (2.67).2.8 Normalized De�nitions of Numerical ErrorsIt is important to have clear, insightful, consistent de�nitions of errors beforemeaningful comparisons can be made. In general, the numerical error for a schemewith grid with spacing h and time step size �� depends on the Courant number� � c��h (2.81)and the grid resolution or number of cells per wavelengthN = h (2.82)where ` is the wavelength of the disturbance. The resolution N is an importantparameter representing the trade-o� between accuracy and cost. Higher resolutionsresult in smaller truncation errors and better solutions. The greater the formal orderof accuracy of a scheme, the faster error drops when N is increased. But higherresolution also increases cost and this cost grows more rapidly with the number ofdimensions. In 1D, the storage space required to resolve a wavelength is linear in N .In 2D, it is quadratic and in 3D, it is cubic. Time step size shrinks inversely with

33N , due to stability constraints, so that the computational time required to solve acertain problem grows one order in N faster than the memory requirement. Therapid rise in cost is the major motivator for �nding ways to improve accuracy whilekeeping N low.In this work, the di�erences between numerical solutions and exact solutions areseparated into two classes: phase speed error and amplitude error. We de�ne thephase error to be the fractional error in propagation speed,�p = cnumcex � 1 (2.83)where cnum is the numerical propagation speed and cex is the exact propagation speed.It is common in numerical electromagnetic to use sources with some speci�edfrequency ! to introduce disturbance into the numerical integration, in which case,the numerical frequency is exact, and the error appears in the numerical wavelengths`num. It is trivial to show that the normalized error (2.83) is also the normalized errorfor the wavelength �p = �` = `num`ex � 1 (2.84)and can be related to the normalized error for the propagation constant (wave num-ber) � by �� = �num�ex � 1 = �p�p + 1 � ��p (2.85)In this dissertation errors will be presented in terms of the propagation speed, butthis can be converted to (or simply taken as an estimate of) the di�erent forms usedin literature from the electromagnetic and acoustic communities.The amplitude error is de�ned to be the fractional reduction in amplitude afterthe wave travels for one period `=cex. A scheme that advances the solution by ��

34each time step requires `c�� = N�time steps to advance it one period, thus the normalized error is de�ned for thisnumber of time steps. The de�nition of amplitude error is�a = j�jN=� � 1 (2.86)where � is the ampli�cation factor (of the physical mode, see Appendix C) of thenumerical scheme. When presented in this dissertation, both �p and �a are in fractionsunless explicitly stated to be percentages.For uniformity of scaling, the \excess ampli�cation" of the spurious modes arede�ned using equation (2.86) with eigenvalue of the spurious mode rather than thephysical mode. The spurious mode does not have an associated exact propagationspeed, but the error is still normalized for the same time, i.e., one period. Note that�a must be negative or the amplitude error (or excess ampli�cation) grows withoutbound. Neutrally stable (non-dissipative) schemes or modes are those with �a = 0.After propagating for m periods, the phase error is o� by a total of m�p timesone cycle; that is, �p;total = 2�m�p (2.87)radians, and the amplitude is o� by a total of�a;total = (1 + �a)m � 1 (2.88)times the correct amplitude. Depending on the application, m can vary greatly orvery little. Figure 2.3 shows the total phase error (2.87) and total amplitude error(2.88) that would be present in a wave after propagating for m periods, for schemeswith some given �p and �a. The normalized errors in Figure 2.3 are �p and �a and the

35total error is the total fractional error incurred after propagating for m periods. It issurprising to see that although phase error accumulates linearly and the amplitudeerror accumulates geometrically, the total is virtually the same, to a total error ofabout 10 percent. Since only small values of �p and �a are of interest (total errors ofgreater than, say, 15 percent are unlikely to be of any use except in a limited subsetof practical numerical simulations), phase and amplitude errors are treated as being\equally bad" in this work. Many of the numerical schemes in this dissertation havezero amplitude error, but they are by no means ideal in every situation. For problemswith more tolerance for one type of error or the other, the comparison remain simpledue to the quantitative similarity of the errors.0.0001

0.001

0.01

0.1

1

1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1

tota

l err

or

normalized error

m=1, phase & amp errm=10, phase errm=10, amplt err

m=100, phase errm=100, amplt err

m=1000, phase errm=1000, amplt err

Figure 2.3: Total error vs. normalized error.Group speed errors are being increasingly recognized as an important character-istic of numerical schemes. This dissertation explicitly presents only the phase speederrors, however, because the group speed errors can be deduced from the phase speederror and the formal order of accuracy of the scheme. In fact, group speed error isvery nearly proportional to the phase speed error. This relationship holds especiallyin moderate and high resolutions where the lowest order error term dominates.

36To show the relationship between phase speed error and group speed error, thenumerical frequency is written as the product!num = cnum�ex (2.89)of the numerical speed and the exact wave number. The numerical propagation speedcan be expanded in the general sumcnum = cex "1 + 1Xi= ai�i# (2.90)where is the order of accuracy of the numerical scheme and� = �exh (2.91)is the exact wave number non-dimensionalized by the grid spacing. The phase speedrelationship cnum = !num� is automatically satis�ed regardless of how cnum is written.One can easily show that the numerical group speed iscg;num = @!num@� = cex "1 + 1Xi=(i+ 1)ai�i# (2.92)The leading error term is ( + 1)ai�, thus for a -th order accurate scheme, thegroup speed approaches +1 times the phase speed error away from extremely highfrequency disturbances. For extremely high frequency disturbances, the large phaseerror may render the solution unusable anyway.

CHAPTER IIISCHEMES IN ONE DIMENSIONIn preparation for multi-dimensional schemes, this chapter explores a multitude of1D schemes upon some, the multi-dimensional schemes are based. Only the advectionscheme is treated in 1D since the 1D versions of Yee's scheme for the linear Maxwell'sequations (2.8), acoustic equations (2.13) and the advection equation (2.79) all yieldidentical numerical and exact behaviors. And of course in 1D, the bicharacteristicequations directly reduce to the advection equations. After examining the schemesin 1D, we will move to multi-dimensional schemes in the next two chapters.Before starting, we note, as shown in Appendix B, that Yee's scheme is equivalentto the standard leapfrog scheme on a grid that is twice as �ne. We will refer to thestandard leapfrog scheme, the central-di�erence leapfrog scheme and Yee's schemesynonymously, but the standard leapfrog scheme is always analyzed at twice theresolution so that its numerical behavior is equivalent to Yee's scheme. The analysesdone in this dissertation for Yee's scheme are actually the equivalent analyses of thestandard leapfrog schemes at twice the resolution.Higher-order extensions have been examined by others such as Deveze [27] whopresented the fourth-order extension. Next, the upwind leapfrog schemes are pre-sented. The second-order accurate upwind leapfrog scheme was �rst described by37

38Iserles [49] and other extensions and applications were made in references [50, 3, 47,48]. The upstream-biased schemes for bicharacteristic equations derived in ChapterII are �rst based on the 1D linear advection equation (2.79) for which the upwindleapfrog schemes were �rst developed. Then the cross terms are added afterward toextend to higher dimensions. We will note the philosophical and practical di�erencesbetween the upstream-biased schemes and the central-di�erence schemes.The standard leapfrog and upwind leapfrog schemes are reversible and thereforehave no spurious numerical damping, i.e., �a = 0, as long as they are stable. Wewill also explore irreversible central-di�erence schemes with which to compare them.Since we have argued that phase speed error is as undesirable as amplitude error,the irreversible schemes serve to put into perspective the virtues and weaknesses ofthe reversible schemes.This chapter is not meant to be a thorough treatment of the types of schemesexamined. We limit ourselves to three-level schemes, since two-level schemes havebeen treated much more thoroughly in the literature. The coverage of three-levelschemes are also by no means complete. We limit ourselves to examples that arereasonably manageable in a computer code. Hence they remain moderately compact,using no more than eight points in each di�erence stencil.3.1 Reversible Central-Di�erence SchemesUsing the standard leapfrog stencil in Figure 3.1 on equation (2.79a), the timederivative is replaced by the di�erences between points (i; n+1) and (i; n�1), and thespace derivative is replaced by the di�erence between points (i+ 12 ; n) and (i� 12; n),resulting in the �nite-di�erence equationGn+1;i �Gn�1;i2�� + cGn;i+1=2 �Gn;i�1=2h = 0 (3.1)

39for the quantity G advected in the direction +x1. Note that in Figure 3.1 points,A and B are separated by �x, while points C and D are separated by 2�� , andthat �� is the time di�erence between any two adjacent levels. The time step sizede�ned here is slightly di�erent than the scheme described by Yee [14], but otherwisethe numerical algorithms are identical. (See Appendix B for a comparison of Yee'sscheme and what is called the standard leapfrog scheme is this dissertation.) We givethis second order Yee's scheme for advection the designation \sl1dAd-O2". The �rstpair of letters stand for general scheme (standard leapfrog), the second pair standsfor the dimension of the problem (1d), third pair stands for the equation being solved(advection) and the last stands for a second-order accurate scheme. This scheme canbe interpreted by a quadratic polynomial interpolation through points A, B and C'to obtain the solution at point D', for a wave that is moving to the right. (Such aconstruction would yield an equivalent discretization.) The solution at points C andC' are identical as are the solution at points D and D', because they lie on the samecharacteristics (dotted lines).The von Neumann analysis [51] (described in Appendix C) of this scheme yieldsthe ampli�cation factors 8>><>>: �2I� sin �2 +q1 � 4�2 sin2 �2�2I� sin �2 �q1� 4�2 sin2 �2 9>>=>>; (3.3)for this scheme. The �rst ampli�cation factor is for the physical mode and the secondfor the spurious mode (which comes from having three time levels). The stabilitylimit can easily be shown to be �max = 12 for this scheme. When the scheme is1The �nite-di�erence equation for the quantity advected in the direction -x isSn+1;i � Sn�1;i2�� � cSn;i+1=2 � Sn;i�1=2h = 0 (3.2)which is identical except for the sign on the second term and that G is replaced by S . The errorfor this equation is the same as that for the +x direction.

40n + 1nn � 1C i+ 12ii� 12c��DA D' C' B x�

Figure 3.1: Standard leapfrog stencil for second-order accurate 1D advection, schemesl1dAd-O2.stable (0 � � � �max), both ampli�cation factors have moduli unity, meaning thatYee's scheme is neutrally stable. The numerical wave speed is determined from theeigenvalue of the physical mode, as described in Appendix C and discussed below.The normalized errors of the physical mode for this scheme are plotted in Fig-ure 3.2. (The contours used in Figure 3.2 and similar �gures in this chapter areinterpreted in the following convention: Solid lines represent the zero-error contour.Thin dot-dashed lines denote negative error, and thick dot-dashed lines denote pos-itive error. The number of dots in a set denotes the log number of magnitude ofthe error, i.e., single dot is �0:1, double dot is �0:01, triple dot is �0:001, etc.The normalization procedure greatly ampli�es round-o� error in the plotting pro-cess, leading to anomalous behaviors near � � 0 and � � 0. These anomalies maybe disregarded.) The (negative) amplitude error shows that the scheme is dampedwhen the stability limit is exceeded. The scheme is nevertheless unstable, becausethe spurious mode is unstable2. Within its stability limit, the scheme is neutrally2The contour plot of the amplitude error in Figure 3.2 appears to be noisy, but in fact, allthe contours represent zero error everywhere except upper right region. This plot displays trivial

41stable (zero amplitude error). As shown in the plots of �p, the standard leapfrogscheme is exact at its stability limit. This can be seen readily in stencil of Figure3.1, where at � = �max, points D and A coincide, leading to an exact interpolation.At � = 0, the interpolation is also exact, but the error does not go to zero. Thisis because the normalizations (2.83) and (2.86) are for error accumulated over oneperiod, and as � ! 0, the number of steps for one period goes to in�nity.Figure 3.3 shows the dispersion curve for this scheme. This is sometimes used inthe literature to show the accuracy of a numerical scheme in duplicating propagationconstants. The exact propagation constant is shown on the horizontal axis and thenumerical propagation constant is shown on the vertical axis. Both are normalizedby the grid spacing, h. In terms of resolution, one can show that N = 2��exh . Curvesfor � = 13, � = 13 , � = 14 and � = 16 are shown in Figure 3.3. The curve for � = 12also represents the exact solution. We would like to present the accuracy of thisscheme in a di�erent form for several reasons. First, plots of dispersion curves applyonly in 1D. In multi-dimensional simulations, the numerical propagation constantis also dependent on the direction of propagation, which introduces one additionalparameter in 2D and two additional parameters in 3D. Representing the variationsdue to these parameters would cause too much clutter on the plot. Second, thedispersion curve does not give clear information in the regions where it is mostcritical. It is interesting to see how the numerical propagation constant behaves forthe higher end of �exh. (We show this behavior with plots such as Figure 3.2.) Butinformation, but we will keep it for uniformity with plots for other schemes. When we show datafor schemes �nite amplitude error, this plot will be useful. Three-level reversible schemes such asthe standard leapfrog and upwind leapfrog schemes can be shown to have ampli�cation factors inpairs that satisfy j�r�sj = 1 (3.4)with �r being for a physical mode and �s for a spurious mode. So if one is damped, the other isconsequently unstable.

42

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Amplitude

nu

theta

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Phase

nu

theta

Amplitude

0.20.4

0.60.8

nu

1

2

3

theta

-0.1

-0.05

0

0.05

0.1

0.20.4

0.60.8

nu

1

2

3

theta

Phase

0.20.4

0.60.8

nu

1

2

3

theta

-0.5

-0.25

0

0.25

0.20.4

0.60.8

nu

1

2

3

theta

Scheme sl1dAd-O2 (3.1) �max = 12Figure 3.2: Error plots for scheme sl1dAd-O2.

430

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3

.

.�exh�numh � = 12 � = 13� = 16Figure 3.3: Dispersion curve of standard leapfrog scheme for 1D advection.what is probably more useful to users are regions of acceptable errors, say, less than10%. The region where the error is less than around 10% shows that all curves beginto coincide and are impossible to separate with the naked eye. Third, the dispersioncurves do not explicitly give any useful number with which one may use to decideon whether to use one scheme or another. Finding such numbers would entail theof measuring the plot. This could be highly imprecise in regions where the curvesseemingly coincide.We propose to present the errors at several carefully chosen discrete values of thescheme's parameters. Table 3.1 shows at several Courant numbers, the numericalvalues of �p for a resolution of N = 4 and the the resolution required to bring �pto one percent and to one degree per cycle (�p = 1360). These error thresholds arechosen to match those used by previous researchers, i.e. [47, 46]. The error valuesare accompanied by its location on a ruler that is drawn to scale throughout thisdissertation. (In multi-dimensional problems, the range of the error values taken overall propagation angles will be given.) Ignoring the special case of � = �max where

44the scheme is exact (except to note that a minimum resolution of N = 2 is alwaysrequired so that aliasing does not occur), this scheme is about 5% to 10% too slow inits phase speed at N = 4. A resolution of up to N = 12:1 is required for one percentphase speed error at the Courant numbers shown in the table.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 r -9.11r -7.98r -6.25r0 12.111.19.642 22.921.118.22Scheme sl1dAd-O2 (3.1) �max = 12Table 3.1: Error ranges for scheme sl1dAd-O2.Below, we will compare the errors of this scheme with that of its upstream-biasedcounterpart, the second-order accurate reversible upwind leapfrog scheme. Presently,we will look at high-order extensions of this scheme.The standard leapfrog scheme sl2dAd-O2 yields the equivalent di�erential equa-tion Gn+1;i�Gn�1;i2�� + cGn;i+1=2�Gn;i�1=2h= G� + cGx + ch2 (1�2�)(1+2�)24 Gxxx +O(h4) (3.5)where the derivatives are evaluated at the grid coordinates (n; i). This shows thatthe di�erence equations di�er from the di�erential equations by terms on the orderof h2. All derivatives in the equivalent di�erential equations are evaluated at theindices (n; i; j; k) used in the equivalent di�erential equations. This point varies fromscheme to scheme and equation to equation, but it should be clear from context

45where the point falls. In 2D, the index k disappears and in 1D, j also disappears.Second-order accurate approximations of the error term in the equivalent di�eren-tial equations proportional to h2 raise the scheme to fourth-order accuracy whereuponit has the equivalent di�erential equationGn+1;i�Gn�1;i�� + cGn;i+1=2�Gn;i�1=2h�ch2 (1�2�)(1+2�)24 �xxxn;i G= G� + cGx � ch4 (3�2�)(1�2�)(1+2�)(3+2�)1920 Gxxxxx +O(h6) (3.6)The � operators are standard second-order accurate �nite-di�erence operators. Forclari�cation, they are de�ned in Appendix D. (These operators represent what wewould call a reasonable and straight forward discretization of the derivatives, butthey are not unique.) This scheme is designated \sl1dAd-O4". Its stencil and othersin this section are shown in Figure 3.4 for comparison. Repeating the high-orderextension procedure one more time yields the sixth-order accurate scheme with theequivalent di�erential equationGn+1;i�Gn�1;i�� + cGn;i+1=2�Gn;i�1=2h�ch2 (1�2�)(1+2�)24 �xxxn;i G + ch4 (3�2�)(1�2�)(1+2�)(3+2�)1920 �xxxxxn;i G= G� + cGx +O(h6) (3.7)This sixth-order scheme is designated \sl1dAd-O6".The normalized errors for these higher-order schemes for 1D advection are plottedin Figures 3.5 and 3.6. Almost all trends that appeared for the second-order schemereappear in high-order schemes (but of course, the higher order schemes are moreaccurate). Despite greater support, both high-order schemes share the same stabilitylimit of their second-order counterparts, which is �max = 12 . This is not surprising

46sl1dAd-O2 sl1dAd-O4 sl1dAd-O6x� Figure 3.4: Reversible central-di�erence stencils for 1D advection.in light of reference [52] which showed that for simple scalar advection schemes, thenumbers of points to either side of the characteristic line through the updated point(point D) cannot di�er by more than one. Within the stability range, the numericalphase speeds are always on the slow (negative) side. Propagation speeds are exactat � = �max, then drop o� as � is lowered.Tables 3.2 and 3.3 show discrete numerical values for �p at N = 4 and theresolution required for �p = 1% or 1360. The improvement over the second-orderscheme is remarkable. The sixth-order scheme achieves less than one percent errorfor N < 4. The price for this is the the additional computations to resolve moreterms in the equivalent di�erential equation.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 r -2.24r -1.94r-1.49r0 4.944.744.422 6.866.576.12Scheme sl1dAd-O4 (3.6) �max = 12Table 3.2: Error ranges for scheme sl1dAd-O4.The procedure for multi-dimensional high-order extensions are similar to those for

47

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Amplitude

nu

theta

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Phase

nu

theta

Amplitude

0.20.4

0.60.8

nu

1

2

3

theta

-0.1

-0.05

0

0.05

0.1

0.20.4

0.60.8

nu

1

2

3

theta

Phase

0.20.4

0.60.8

nu

1

2

3

theta

-0.4

-0.2

0

0.2

0.20.4

0.60.8

nu

1

2

3

theta

Scheme sl1dAd-O4 (3.6) �max = 12Figure 3.5: Error plots for scheme sl1dAd-O4.

48

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Amplitude

nu

theta

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Phase

nu

theta

Amplitude

0.20.4

0.60.8

nu

1

2

3

theta

-0.1

-0.05

0

0.05

0.1

0.20.4

0.60.8

nu

1

2

3

theta

Phase

0.20.4

0.60.8

nu

1

2

3

theta

-0.4

-0.2

0

0.20.4

0.60.8

nu

1

2

3

theta

Scheme sl1dAd-O6 (3.7) �max = 12Figure 3.6: Error plots for scheme sl1dAd-O6.

49��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 r-0.703r-0.606r-0.464r0 3.753.653.492 4.734.64.372Scheme sl1dAd-O6 (3.7) �max = 12Table 3.3: Error ranges for scheme sl1dAd-O6.1D, but in higher dimensions, the error terms include multi-dimensional derivativesand are more numerous. In Chapter V, we treat the multi-dimensional equations foracoustics and electromagnetics.3.2 Reversible Upstream-biased SchemesThe central-di�erence schemes are derived by making the best possible approxi-mation of the derivatives given the data close to and balanced around the point atwhich the derivative must be evaluated. One problem with this is that in the courseof the following time step, all variables get advected downstream and the balance isthrown o�. This is more obvious in the interpretation the central-di�erence stencilof Figure 3.1 as a quadratic interpolation for point D' from points A, C' and B. Fromthe perspective of a point on the characteristic through D' where the new solutionis taken, the interpolation is poorly balanced, using too much data too far to theright of D'. From a stationary perspective centered on the x-coordinate of D, thedetermination of the new solution is using too much information that has gone by(C' and B) and not enough of the information that is approaching in the currenttime step (A). This is typical of central-di�erence schemes for hyperbolic equations

50and is roughly compensated for in upstream-biased discretizations.Upstream-biased schemes look at the direction and sense of the traveling wavesin addition to approximating the partial di�erential equation. More weight is givento information from the \upstream" side because during the course of the time step,that information moves closer to the point where the derivative is sought. As theupstream points get advected downstream in the course of the time step, they becomemore balanced around the interpolated point. Consequently, the �rst noticeabledi�erence between the central-di�erence schemes and the upstream-biased schemesis that the upstream-biased schemes use stencils that do not necessarily contain theclosest possible points around the point being updated.Our �rst upwind leapfrog scheme uses the stencil in Figure 3.7 on equation(2.79a). The time derivative is replaced by an average of di�erences between points(i+ 12 ; n+1) and (i+ 12; n) and points (i� 12; n) and (i� 12; n�1), and the space deriva-tive is replaced by the di�erence between points (i+ 12 ; n) and (i� 12 ; n), resulting inthe di�erence equationGn+1;i+1=2 �Gn;i+1=2 +Gn;i�1=2 �Gn�1;i�1=22�� + cGn;i+1=2 �Gn;i�1=2h = 0 (3.8)for G. We give this second-order upwind leapfrog scheme for 1D advection thedesignation \ul1dAd-O2", where the �rst two letters stand for \upwind leapfrog".As shown in Figure 3.7, this scheme is equivalent to using a quadratic polynomialinterpolation through points A, B and C' for the solution at point D'. The problemwith the central-di�erence schemes earlier in this section is alleviated here, since thestencil is biased toward the left side3. The points used in interpolation are more3If the wave is coming from the right, such as in equation (2.79b), the bias would be to the right,and the di�erence equation would beSn+1;i�1=2 � Sn;i�1=2 + Sn;i+1=2 � Sn�1;i+1=22�� � cSn;i+1=2 � Sn;i�1=2h = 0 (3.9)

51nn + 1C n � 1i+ 12i� 12 c��

DA D' C' B x�Figure 3.7: Upwind leapfrog stencil for second-order accurate 1D advection, schemeul1dAd-O2.balanced around the point D'. At � = 12, they are perfectly balanced.The von Neumann analysis of this scheme yields ampli�cation factors8>><>>: (12 � �)(1 � e�I�)�qe�I� + (12 � �)2(1 � e�I�)2(12 � �)(1 � e�I�) +qe�I� + (12 � �)2(1� e�I�)2 9>>=>>; (3.10)the �rst of which is for the physical mode and the second for the spurious mode.One can show exactly that for 0 � � � �max = 1, both ampli�cation factors havemoduli unity, meaning that the upwind leapfrog scheme is neutrally stable. Note thatthe upwind leapfrog scheme has a higher stability limit than the standard leapfrogscheme, even though they share the same amount of spatial support of one cell. Thedi�erence is that the upwind leapfrog scheme has a wider support in its zone ofdependence. This is consistent with the CFL condition [53].The normalized phase speed error is shown in Figure 3.8 for the upwind leapfrogscheme. Note that �a = 0 for 0 � � � 1. The phase speed is exact at its stabilitylimit of �max = 1 and again at half this number. This can be seen readily in Figure3.7 where at � = 12�max, points D' and C' coincide, making the interpolation exact.

52This is a very useful property when the ratio of indices of refraction in the sameproblem is close to two. In contrast, scheme sl1dAd-O2 is exact only at its stabilitylimit. Unfortunately, in higher dimensions, the upper half of the stability range ofthe upwind leapfrog scheme becomes unstable and only the lower half is usable.The �p plots in Figure 3.8 show that the second-order upwind leapfrog schemeul1dAd-O2 has improved phase speed errors over the second-order standard leapfrogscheme sl1dAd-O2. But as � gets very low, the standard leapfrog scheme degradesmore gracefully than does the upwind leapfrog scheme. Note the di�erences in slopesof the �p contours between Figures 3.2 and 3.8. While it may be assumed that anyscheme would be run in the upper range of its stability limit to save computationalcost, this unexpected behavior is very interesting and can occur in practice if indicesof refraction vary greatly. It is hypothesized that at low Courant numbers, the awsof the central-di�erence scheme, as brought up earlier in this section, actually becomeits strength. As � ! 0, the unbalance of the standard leapfrog scheme vanishes. Forthe upwind leapfrog scheme, the upstream-biasing, aimed at restoring balance for themovement of the waves, \over compensates". As � ! 0, the upwind leapfrog schemebecomes the most unbalanced. This continues to hold for higher-order versions ofthese schemes. The irreversible upstream-biased schemes which will be presentedbelow can give a more exible amount of compensation.Numerical values of �p for N = 4 and the resolution required for �p = 1% is givenin Table 3.4. The ul1dAd-O2 scheme is slow for 0 < � < 12 and fast for 12 < � < 1. Attwo-thirds their respective stability limits, the upwind leapfrog scheme has roughlyone-third the error of the standard leapfrog scheme, although it should be noted thatthe stability limit is twice as high for ul1dAd-O2.The second order upwind leapfrog scheme, ul1dAd-O2 (3.8) yields the equivalent

53

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Amplitude

nu

theta

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Phase

nu

theta

Amplitude

0.20.4

0.60.8

nu

1

2

3

theta

-0.1

-0.05

0

0.05

0.1

0.20.4

0.60.8

nu

1

2

3

theta

Phase

0.20.4

0.60.8

nu

1

2

3

theta

-0.1

0

0.1

0.2

0.3

0.20.4

0.60.8

nu

1

2

3

theta

ul1dAd-O2 (3.8), �max = 1Figure 3.8: Error plots for scheme ul1dAd-O2.

54��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 r4.56r0r -2.28r0 8.5526.042 16.2211.52ul1dAd-O2 (3.8), �max = 1Table 3.4: Error ranges for scheme ul1dAd-O2.di�erential equationGn+1;i+1=2�Gn;i+1=2+Gn;i�1=2�Gn�1;i�1=22�� + cGn;i+1=2�Gn;i�1=2h= G� + cGx � ch2 (1��)(1�2�)12 Gxxx +O(h4) (3.11)showing that the di�erence equations di�er from the di�erential equations by termson the order of h2. As in the central-di�erence schemes, it can be extended tofourth-order accuracy, resulting in the equivalent di�erential equationGn+1;i+1=2�Gn;i+1=2+Gn;i�1=2�Gn�1;i�1=22�� + cGn;i+1=2�Gn;i�1=2h+ch2 (1��)(1�2�)12 �xxxn;i G= G� + cGx + ch4 (1��)(2��)(1+�)(1�2�)240 Gxxxxx +O(h6) (3.12)and sixth-order accuracy, resulting in the equivalent di�erential equationGn+1;i+1=2�Gn;i+1=2+Gn;i�1=2�Gn�1;i�1=22�� + cGn;i+1=2�Gn;i�1=2h+ch2 (1��)(1�2�)12 �xxxn;i G � ch4 (1��)(2��)(1+�)(1�2�)240 �xxxxxn;i G= G� + cGx +O(h6) (3.13)The stencils for these schemes are extended in space and shown in Figure 3.9. (Ad-ditionally, extensions in time are possible [47, 50].)

55ul1dAd-O2 ul1dAd-O4 ul1dAd-O6x� Figure 3.9: Reversible upstream-biased stencils for 1D advection.The normalized errors for these higher-order schemes for 1D advection are com-puted and plotted in Figures 3.10 and 3.11 and numerical values are shown in Tables3.5 and 3.6. Almost all trends that appeared for the second order upwind leapfrogscheme reappear in these high-order schemes. As in the central-di�erence schemesgreater support did not lead to greater stability ranges. The stability limits for allupwind leapfrog schemes are �max = 1. Within the stability range, the numericalphase speeds are always on the slow (negative) side for 0 < � < 12 and on the fast(positive) side for 12 < � < 1. For � = 12 and � = 1, all upwind leapfrog schemes areexact.In the upper half of the stability ranges for the upwind leapfrog schemes, they aremore accurate than the standard leapfrog schemes are. However, one can see fromthe slopes of the �p contours that the standard leapfrog schemes still degrade moregracefully than do the upwind leapfrog schemes at lower Courant numbers. In fact,by � = 0, the degradation has leveled out for the standard leapfrog schemes, but notfor the upwind leapfrog schemes.Of the schemes with the same order of accuracy, the upwind leapfrog schemeshave roughly one-third the error of the standard leapfrog schemes at � = 23�max, andone-half the error at � = 13�max. This was the trend observed with the second orderschemes.Although these schemes do very well toward their respective �max, they all degrade

56

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Amplitude

nu

theta

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Phase

nu

theta

Amplitude

0.20.4

0.60.8

nu

1

2

3

theta

-0.1

-0.05

0

0.05

0.1

0.20.4

0.60.8

nu

1

2

3

theta

Phase

0.20.4

0.60.8

nu

1

2

3

theta

-0.02

0

0.02

0.04

0.20.4

0.60.8

nu

1

2

3

theta

ul1dAd-O4 (3.12) �max = 1Figure 3.10: Error plots for scheme ul1dAd-O4

57

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Amplitude

nu

theta

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Phase

nu

theta

Amplitude

0.20.4

0.60.8

nu

1

2

3

theta

-0.1

-0.05

0

0.05

0.1

0.20.4

0.60.8

nu

1

2

3

theta

Phase

0.20.4

0.60.8

nu

1

2

3

theta

-0.02

0

0.02

0.04

0.20.4

0.60.8

nu

1

2

3

theta

ul1dAd-O6 (3.13) �max = 1Figure 3.11: Error plots for scheme ul1dAd-O6

58away from it. Absolute �p at � = 23�max are about half that at � = 13�max. At� = 13�max, only the sixth-order schemes get less than one percent error for N = 4.This is not to say that the high-order schemes are not worth their cost. They convergemuch faster as N increases.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 r1.12r0r-0.559r0 4.1223.422 5.7424.812ul1dAd-O4 (3.12) �max = 1Table 3.5: Error ranges for scheme ul1dAd-O4.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 r0.351r0r-0.176r0 3.2922.872 4.1723.672ul1dAd-O6 (3.13) �max = 1Table 3.6: Error ranges for scheme ul1dAd-O6.Additional high-order leapfrog schemes in 1D can be found in reference [50], andin Section 3.6 below.

593.3 ReversibilityThe leapfrog schemes presented so far are reversible schemes. That is, the samediscrete equation used in advancing in time can also be used to back-track andexactly recover the previous solution. In the central-di�erence leapfrog schemes,the point updated and the point recovered have the same spatial coordinates, andin upstream-biased leapfrog schemes, the updated point is on the downstream sidewhile the recovered point is on the upstream side, consistent with moving informationin reverse when the sense of time is reversed. Both the central-di�erence and theupstream-biased schemes can be written in the general formXl;i;j;k;nC l(i; j; k; n)U lio+i;jo+j;ko+k;no+n = 0 (3.14)For a system of equations, the superscript indicates the l-th variable in the system.The indices io, jo, ko and no are the coordinates of a location �xed to the stencil.C l(i; j; k; n) is the coe�cient corresponding to the term displaced by indices (i; j; k; n)from the the point (io; jo; ko; no). And U lio+i;jo+j;ko+k;no+n is the (known or unknown)solution of the l-th variable at the point (io + il; jo + jl; ko + kl; no + nl).Scheme (3.14) is reversible for a wave propagation equation if it is antisymmetricabout the center of the space-time stencil (or weights of the points directly oppositeeach other, with respect to the center of symmetry, have opposite signs). In otherwords, for (io; jo; ko; no) at the point of geometric symmetry, each of its coe�cientsC(i; j; k; n) satis�es the conditionC(i; j; k; n) = �C(�i;�j;�k;�n) (3.15)To advance the solution, equation (3.14) is solved for the variable(s) at the latesttime step. To back-track, it is solved for the variable(s) at the earliest time step.

60The von Neumann ampli�cation factors are the eigenvalues of the resulting systemof discrete equations. For reversible schemes, the discrete equations are the same formoving the solution forward or backward. Their ampli�cation factors are also similar,with the factor for the backward direction equal to the inverse of the correspondingfactor in the forward direction. If the scheme is stable in the forward direction, it musthave no ampli�cation factor with modulus greater than unity, and the inverse of everyampli�cation factor also must be no greater than unity for the backward directionto be stable. Hence, if a reversible scheme is stable, all ampli�cation factors musthave moduli unity, and the scheme must be neutrally stable and free of dissipation.Without dissipation, the only error incurred by reversible schemes are errors in thepropagation speed. Non-dissipative schemes are desirable in many situations, evenwith �nite error in propagation speeds. For long integration times where dissipativeschemes leave little signal to be detected, non-dissipative schemes at least leave thefull wave strengths, even if the phase may be wrong. In some classes of practicalmonochromatic problems where each wave spends roughly the same amount of timein the computational domain and only a small number of periods of the end result isrequired, the error accumulated over a long integration time in e�ect results only in aphase shift that does not signi�cantly a�ect the transformation to far-�eld scatteringpatterns.However, there are many problems in which dispersive errors can have suddenand pronounced e�ects. These are problems with multi-bounce (such as in cavities,resonators and inlets), cancellations and additions (such as RCS near a ground plane).The results of wave superpositions can be signi�cantly a�ected if some waves arriveslightly shifted. For example, the location of peaks and nulls may be poorly predicted.The non-dissipative schemes, such as leapfrog, Yee's scheme [14] and the family

61of upwind leap-frog schemes, including those in [47, 54, 3, 55, 48], have no amplitudeerror. In the context of reversible upstream-biased schemes, many variations havebeen used to reduce the �nite phase speed error [54, 47].While having no spurious dissipation is ideal, these non-dissipative schemes canstill be improved upon. Without amplitude error, the phase speed error amounts tothe worst error, so it is natural to seek to reduce it at the cost of \a little dissipation".In addition, as seen in this chapter and in Chapters IV and V, at low Courant num-bers, the upwind leapfrog schemes do not degrade as gracefully as do the standardleapfrog schemes when � is low. Allowing dissipation gives extra degrees of freedomto improve the accuracy of the upstream-biased schemes at low Courant numbers.3.4 Derivation of 1D Linear SchemesFor any of the stencils presented in this chapter, including the reversible ones, thenumerical scheme can be derived by equating the sum of the variables in the stencilwith the advection equation, i.e.,Xi;n 'i;nGio+i;no+n = G� + cGx (3.16)where ' is a weighting factor. The dependent variables on the left hand side arereplaced by Taylor series expansion and similar terms are equated. Speci�cally, thecoe�cients of h0, h2, h3, etc. and of ut and ux are used. The last two correspond toterms of order h. The system of algebraic equations resulting from these restrictionsis solved for the coe�cients ' of the discrete variables.In a di�erent approach, the updates for the irreversible schemes can also be foundby the polynomial interpolations described above. This leads to the same results asequating coe�cients.

62In the reversible schemes presented, no errors term contain even-ordered deriva-tives, another indication that they are free of dissipation. Odd-ordered derivatives inthe truncation error indicates spurious dispersion and even-ordered derivatives indi-cate spurious damping. The appearance of even-ordered derivatives in the equivalentdi�erential equations indicate that the schemes are dissipative.3.5 Irreversible Central-Di�erence SchemesTwo irreversible central-di�erence schemes are presented. These use the stencilsshown in Figure 3.12. The �rst, designated \ciA" (the �rst letter \c" stands forcentral-di�erence, the second letter \i" for irreversible and the third letter \A" forthe individual scheme in the class) has equivalent di�erential equation(2��)(2+�)(6��(1+�2))Gn+1;i + (2��)(2+�)(�1+2�)(1+2�)(6��(1+�2)) Gn;i+(�2+�)(�1+�)(�1+2�)(12��(1+�2)) Gn�1;i+1 � (1+�)(2+�)(1+2�)(12��(1+�2)) Gn�1;i�1+(�2+�)(1+�)(�1+2�)(1+2�)(12��(1+�2)) Gn;i�1 + (�1+�)(2+�)(�1+2�)(1+2�)(12��(1+�2)) Gn;i+1 =G� + cGx+ch4 (�4+21�2�21�4+4�6)(720(1+�2)) Gxxxxx + ch5 �(�4+21�2�21�4+4�6)(4320(1+�2)) Gxxxxxx+ch6 (�4+21�2�21�4+4�6)15120 Gxxxxxxx + ch7 �(�8+38�2�21�4�13�6+4�8)(120960(1+�2)) Gxxxxxxxx+O(h8) (3.17)and its errors are shown in Table 3.7. As revealed by the truncation error, thisscheme is fourth-order accurate in �p and �fth-order in �a. The second, designated\ciB" has equivalent di�erential equationGn+1;i (3��)(3+�)5��(3+2�2) +Gn;i (�3+�)(�2+�)(2+�)(3+�)(�1+2�)(1+2�)20��(3+2�2)Gn�1;i+1 (�3+�)(�2+�)(�1+2�)10��(3+2�2) �Gn�1;i�1 (2+�)(3+�)(1+2�10��(3+2�2) +

63Gn;i+2 (�2+�)�(1+�)(3+�)(�1+2�)(1+2�)120��(3+2�2) +Gn;i�2 (�3+�)(�1+�)�(2+�)(�1+2�)(1+2�)120��(3+2�2) +Gn;i+1 (1��)(�3+�)(2+�)(3+�)(�1+2�)(1+2�)30��(3+2�2) +Gn;i�1 (2��)(�3+�)(1+�)(3+�)(�1+2�)(1+2�)30��(3+2�2) =G� + cGx+ch6 36�193�2+210�4�57�6+4�825200(3+2�2) Gxxxxxxx+ch7 �(36�193�2+210�4�57�6+4�8)201600(3+2�2) Gxxxxxxxx +O(h8) (3.18)and its errors are shown in Table 3.8. This scheme is sixth-order accurate in �p andseventh-order in �a. Both schemes are stable with stability limits of �max = 12 . TheciA ciBx�Figure 3.12: Irreversible central-di�erence stencils for 1D advection.center points for the earliest time level are conspicuously missing from these schemes.Including these points results in an unstable new scheme.Plots of the amplitude and phase errors for these schemes are shown in Figures3.13 and 3.14. The phase speed error of these two schemes are on the slow side,except in a small region of extremely high frequency and � near the stability limit.The boundary between regions of slow and fast phase errors, de�nes a relationshipbetween � and � for which �p = 0, but this is not at all useful. Both phase andamplitude errors degrade as � is lowered, but unlike all other 1D schemes presentedin this chapter, these degradations are reversed for low enough nu, and near � = 0,errp returns to zero (but not erra). One possible use for these schemes is as a part ofa hybrid solver that switches to these schemes when c (and consequently, �) is very

64

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Amplitude

nu

theta

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Phase

nu

theta

Amplitude

0.20.4

0.60.8

nu

1

2

3

theta

-0.1

-0.05

0

0.05

0.1

0.20.4

0.60.8

nu

1

2

3

theta

Phase

0.20.4

0.60.8

nu

1

2

3

theta

-0.04

-0.02

0

0.02

0.04

0.20.4

0.60.8

nu

1

2

3

theta

ciA (3.17) �max = 12Figure 3.13: Error plots for scheme ciA.

65

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Amplitude

nu

theta

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Phase

nu

theta

Amplitude

0.20.4

0.60.8

nu

1

2

3

theta

-0.1

-0.05

0

0.05

0.1

0.20.4

0.60.8

nu

1

2

3

theta

Phase

0.20.4

0.60.8

nu

1

2

3

theta

-0.01

0

0.01

0.20.4

0.60.8

nu

1

2

3

theta

ciB (3.18) �max = 12Figure 3.14: Error plots for scheme ciB.

66��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 r -2.87r -1.79r-0.977r0 5.234.733.972 7.196.645.832��max 100�a(N = 4) N(�a = 1%) N(�a = 1360)1312231 r -9.84r -8.9r -6.18r0 6.176.195.862 7.887.977.582ciA (3.17) �max = 12Table 3.7: Error ranges for scheme ciA.small. Even when c changes sign, these schemes still work. This may be desirablein more complex multi-dimensional advection problems. The amplitude error showsclearly that the schemes are unstable for � > 12.Comparison of �p between ciA and the fourth-order accurate standard leapfrogscheme at N = 4 shows that they are comparable, with each being somewhat betterin di�erent regions of the stability range. Comparison of �p between ciB and thesixth-order accurate standard leapfrog scheme at N = 4 shows that the reversiblescheme is slightly more accurate. Although the stencils show that scheme ciA isarguably similar to ciB in the same way that sl1dAd-O4 is similar to sl1dAd-O6,there is no clear correlation regarding whether the reversible scheme has better orworse phase speed error.

67��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 r-0.604r-0.374r-0.204r0 3.653.2322 4.574.243.742��max 100�a(N = 4) N(�a = 1%) N(�a = 1360)1312231 r -2.18r -1.92r-1.29r0 4.444.394.162 5.265.255.022ciB (3.18) �max = 12Table 3.8: Error ranges for scheme ciB.Of course, with irreversible schemes, phase speed error is not the whole story.The damping errors for both ciA and ciB are both several times greater than theirphase speed error at N = 4. In treating both types of error equally, one would notprefer these irreversible schemes over the reversible ones (they are even more com-putationally costly in comparison), except perhaps for their more compact stencils.Although the irreversible central-di�erence schemes are not clearly more desirablethan the standard leapfrog schemes, we show in the next section that the use ofirreversible schemes in combination with upstream-biasing can produce arguablymore desirable schemes.

683.6 Irreversible Upstream-biased SchemesIrreversible upstream-biased schemes are similar to upwind leapfrog schemes.Four irreversible upstream-biased schemes are presented, using the stencils shownin Figure 3.15. There is no set guideline for selecting these stencils, except thatwhen the points on the stencils are projected along their characteristic lines to onetime level, they are somewhat balanced about the point being updated.The �rst irreversible upstream-biased scheme is designated \uiA". (The �rst letterstands for upstream-biased and the second for irreversible and the third for theindividual scheme in the class.) It has equivalent di�erential equation1�2�3�� Gn;i�1 � 1+�3�� Gn�1;i�1 + 1+�6���Gn+1;i+(�1+�)(�1+2�)6��� Gn�1;i + (1+�)(�1+2�)3��� Gn;i =G� + cGx+ch3 1�2���2+2�372 Gxxxx + ch4�2+5��5�3+2�4360 Gxxxxx+ch5 3�9�+5�2+5�3�8�4+4�52160 Gxxxxxx + ch6�2+7��7�2+7�4�7�5+2�67560 Gxxxxxxx+ch7 5�20�+28�2�14�3�14�4+28�5�19�6+6�7120960 Gxxxxxxxx +O(h8) (3.19)and its errors shown in Table 3.9. This scheme is third-order accurate in �a andfourth-order accurate in �p. The second, designated \uiB", has equivalent di�erentialequation (�2+�)(2+�)3��(3��) Gn�1;i�1 + (1+�)(2+�)6��(3��)� Gn+1;i+(�2+�)(�1+�)(�1+2�)6��(�3+�)� Gn�1;i + (�1+�)�(�1+2�)6��(�3+�) Gn;i�2+(�2+�)(2+�)(�1+2�)3��(3��) Gn;i�1 + (�2+�)(1+�)(2+�)(�1+2�)6��(�3+�)� Gn;i =G� + cGx+ch4�4+8�+5�2�10�3��4+2�5360(�3+�) Gxxxxx + ch5 16�36��12�2+45�3�6�4�9�5+2�62160(�3+�) Gxxxxxx+

69ch6�44+108�+7�2�119�3+49�4+7�5�12�6+4�715120(�3+�) Gxxxxxxx+ch7 52�136�+19�2+126�3�84�4+21�5+11�6�11�7+2�8(60480(�3+�)) Gxxxxxxxx +O(h8) (3.20)and its errors are shown in Table 3.10. This scheme is fourth-order accurate in �a and�fth-order accurate in �p. The third, designated \uiC", has equivalent di�erentialequation 2��6���Gn+1;i � 1+�3�� Gn�1;i�1 + (�2+�)(�1+2�)6�� Gn;i�1+(�2+�)(�1+2�)6��� Gn�1;i + �(�1+2�)6�� Gn;i+1 + (2��)(1+�)(�1+2�)3��� Gn;i =G� + cGx+ch4�2+5��5�3+2�4360 Gxxxxx + ch5 (�1+�)2(2�3��3�2+2�3)2160 Gxxxxxx+ch6�2+7��7�2+7�4�7�5+2�67560 Gxxxxxxx + ch7 (�1+�)2(2�5�+2�2+2�3�5�4+2�5)60480 Gxxxxxxxx+O(h8) (3.21)and its errors are shown in Table 3.11. This scheme is also fourth-order accuratein �a and �fth-order accurate in �p. The fourth, designated \uiD", has equivalentdi�erential equation2(�2+�)(1+�)9�� Gn�1;i�1 + (�2+�)(1+�)18��(�1+�)�Gn+1;i + 2(�2+�)(�1+2�)9��(1��) Gn;i�1+(1��)(�2+�)(�1+2�)18��� Gn�1;i + 2(1+�)(�1+2�)9��� Gn;i + �(1+�)(�1+2�)18��(1��) Gn�1;i�2 =G� + cGx+ch4 (1��)(2�3��3�2+2�3)540 Gxxxxx + ch5 (1��)(�2+�)2(�1+�+2�2)1620 Gxxxxxx+ch6 (1��)(22�55�+8�2+43�3�34�4+8�5)22680 Gxxxxxxx+ch7 (1��)(�2+�)2(�13+26�+7�2�20�3+12�4)181440 Gxxxxxxxx +O(h8) (3.22)and its errors are shown in Table 3.12.The amplitude and phase errors of these schemes are plotted in Figures 3.16,3.17, 3.18 and 3.19. Of these, schemes uiA and uiC have very bad amplitude errors

70uiA uiBuiC uiDx�Figure 3.15: Irreversible upstream-biased stencils for 1D advection.but uiB and uiD have �a comparable to �p and possible merits. Note the similaritybetween the stencils of schemes ul2dAd-O2 and uiA. The latter has an extra pointbelow the point being updated. This additional point has a remarkable e�ect onthe phase speed error, making �p of scheme ciA �ve times less than �p of schemeul2dAd-O2. But �a is very bad for scheme uiA. At � = 23�max, it is already morethan eight percent.Scheme uiC has one additional point over scheme uiA, on the downstream side.This increases the order of accuracy by one, but speci�cally, it changes �a from O(h3)to O(h5) but leaves �p at O(h4). This reduces �a to a third of the value for schemeuiA and leaves �a is still quite poor. The phase speed error degraded slightly, buthas completely changed direction, being slow for uiC and fast for uiA.Scheme uiB also di�ers from scheme uiA by one additional point, and changesthe order of accuracy in the same manner, but the point is on the upstream side,which makes the comparison between uiB and uiC quite interesting. While �p is slowfor uiC, it is fast for uiB, like uiA. Unlike uiC, the �p of uiB remains fast, like thatof uiA. The amplitude error of uiC is vastly improved over that of uiA and uiB. In

71

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Amplitude

nu

theta

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Phase

nu

theta

Amplitude

0.20.4

0.60.8

nu

1

2

3

theta

-0.1

-0.05

0

0.05

0.1

0.20.4

0.60.8

nu

1

2

3

theta

Phase

0.20.4

0.60.8

nu

1

2

3

theta

-0.02

0

0.02

0.04

0.20.4

0.60.8

nu

1

2

3

theta

uiA (3.19) �max = 12Figure 3.16: Error plots for scheme uiA.

72

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Amplitude

nu

theta

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Phase

nu

theta

Amplitude

0.20.4

0.60.8

nu

1

2

3

theta

-0.1

-0.05

0

0.05

0.1

0.20.4

0.60.8

nu

1

2

3

theta

Phase

0.20.4

0.60.8

nu

1

2

3

theta

-0.05

-0.025

0

0.025

0.05

0.20.4

0.60.8

nu

1

2

3

theta

uiB (3.20) �max = 12Figure 3.17: Error plots for scheme uiB.

73

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Amplitude

nu

theta

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Phase

nu

theta

Amplitude

0.20.4

0.60.8

nu

1

2

3

theta

-0.1

-0.05

0

0.05

0.1

0.20.4

0.60.8

nu

1

2

3

theta

Phase

0.20.4

0.60.8

nu

1

2

3

theta

-0.02

0

0.02

0.20.4

0.60.8

nu

1

2

3

theta

uiC (3.21) �max = 12Figure 3.18: Error plots for scheme uiC.

74

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Amplitude

nu

theta

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Phase

nu

theta

Amplitude

0.20.4

0.60.8

nu

1

2

3

theta

-0.1

-0.05

0

0.05

0.1

0.20.4

0.60.8

nu

1

2

3

theta

Phase

0.20.4

0.60.8

nu

1

2

3

theta

-0.02

0

0.02

0.04

0.20.4

0.60.8

nu

1

2

3

theta

uiD (3.22) �max = 1Figure 3.19: Error plots for scheme uiD.

75��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 r0.927r0.688r0.446r0 3.913.593.192 5.635.154.552��max 100�a(N = 4) N(�a = 1%) N(�a = 1360)1312231r -17r -12.7r -8.35r0 11.19.938.512 17.115.313.12uiA (3.19) �max = 12Table 3.9: Error ranges for scheme uiA.fact, it is less than one percent for N = 4 and � down to 13�max.The additional point in scheme uiB shifts the balance of the interpolation inupstream with respect to uiA, and makes �p faster. In uiC, the additional pointshifts the balance of the interpolation downstream and makes �p slower. In theprevious sections, the central-di�erence schemes were too slow and the upstream-biased schemes were too fast (for 0 < � < 12). The trend (for decreasing �) appearsto be that more upstream-biasing tends to bring in information faster, to make thewaves propagate faster. This suggests that an optimal scheme can be found in somemoderate form of upstream-biasing. (This argument does not work for improving �abecause all schemes with �a > 0 are rejected for stability reasons.)Scheme uiD is one that attempts to make the interpolation for the update points

76��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 r1.38r1.01r0.645r0 4.354.013.562 6.065.64.992��max 100�a(N = 4) N(�a = 1%) N(�a = 1360)1312231 r-0.969r-0.769r-0.515r0 3.983.83.52 5.014.864.522uiB (3.20) �max = 12Table 3.10: Error ranges for scheme uiB.using a highly balanced and compact distribution. All the points used are containedin the region de�ned by the characteristics for � = 0 and � = 1 through the up-dated point. This scheme is stable with �max = 1. It is an enormously successfulscheme using a compact two-cell-wide, six-point stencil, requiring only N = 3:37for both �p<�1% and �a<�1%. This is better than any of the previous schemes usingthe same number of points (including sl2dAd-O4, ul2dAc-O4, ciA, uiB and uiC). Ofthe schemes using eight-point stencils (including sl1dAd-O6, ul1dAc-O6 and ciB),only the wide-stenciled ul1dAd-O6 is more accurate, requiring N = 3:29 cells perwavelength for 1% error.

77��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 r-0.942r-0.718r-0.482r0 3.923.583.122 5.895.394.752��max 100�a(N = 4) N(�a = 1%) N(�a = 1360)1312231 r -6.11r -4.44r -2.79r0 6.245.715.082 8.327.66.732uiC (3.21) �max = 12Table 3.11: Error ranges for scheme uiC.3.7 An Optimized Irreversible Upstream-biased SchemeIntroducing a new stencil (Figure 3.20) for the advection equation (2.79a), wewrite the general linear discrete equationwi;n+1Ui;n+1 + wi�2;nUi�2;n + wi�1;nUi�1;n + wi;nUi;n + wi+1;nUi+1;n+wi�1;n�1Ui�1;n�1 + wUi;n�1 = ut + cux +Xi �ihi (3.23)and match the coe�cients of the lowest-order terms, keeping the weight w of Ui;n�1general. Speci�cally, the coe�cients of h0, h2, h3, h4, ut and ux are used. The lasttwo correspond to terms proportional to h. Resolving these coe�cients, a fourth-order scheme is produced. This formulation is motivated several ways. First, it isa step to sacri�ce perfect amplitude error of the upwind leapfrog scheme to obtain

78��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 r0.518r0r-0.259r0 3.3722.82 4.723.932��max 100�a(N = 4) N(�a = 1%) N(�a = 1360)1312231 r-0.0946r0r-0.0473r0 2.7722.472 3.3923.042uiD (3.22) �max = 1Table 3.12: Error ranges for scheme uiD. exibility in design. Second, by introducing the free parameter w, the scheme maybe adjusted to obtain a family of schemes. Third, based on the hypothesis that theupwind leapfrog schemes become less balanced in the lower half of their stabilityranges, it is anticipated that increasing w as � is lowered will improve the balance.This general scheme is designated \wl" for \weighted leg".The equivalent di�erential equation for scheme wl is�2+��4w(2��(�2+�))Gn+1;i + w��Gn�1;i + 2���2�w2��(�2+�)Gn�1;i�1+(�1+�)�(�2+5��2�2+6�w)12��(�2+�) Gn;i�2 + (1+�)(�2+5��2�2+(6��8)w)4�� Gn;i+2�3��3�2+2�3�(1+3�)2�w4�� Gn;i�1 + �(�2+7��7�2+2�3+(3��)6�w)12��(�2+�) Gn;i+1 =G� + cGx+

79��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 r1.25r0.86r0.518r0 4.233.853.372 5.875.354.72��max 100�a(N = 4) N(�a = 1%) N(�a = 1360)1312231 r-0.439r-0.232r-0.0946r0 3.533.22.772 4.283.893.392uiD (3.22) �max = 12 (stable to � = 1)Table 3.13: Error ranges for scheme uiE.ch4 (�1+�2)(�2+5��2�2+10�w)240 Gxxxxx + ch5 (�1+�2)(6�15�+6�2+(�22�+4�2)w)1440 Gxxxxxx+ch6 (�1+�2)(�16+42��23�2+7�3�2�4+(56��14�2+14�3)w)10080 Gxxxxxxx+ch7 (�1+�2)(18�49�+32�2�14�3+4�4+(�58�+20�2�20�3+4�4)w)40320 Gxxxxxxxx (3.24)For a general w, this is a fourth-order accurate scheme.3.7.1 The Fourth-Order ul1dAd-O4 Scheme: w = 0For w = w4(�) = 0, the point (n � 1; i) drops from the stencil, the odd-ordered(in h) error terms drop out and the fourth-order upwind leapfrog scheme (ul1dAd-O4) from Section 3.2 is recovered. The errors for this scheme were shown previouslyin Figure 3.10. This scheme is clearly stable for 0 � � � 1, but for now, we areconcerned with its performance for 0 � � � 12, because, as shown in Figure 3.10, this

80i� 2 n+ 1nn� 1i+ 1ii� 1Figure 3.20: Stencil for wl schemes.range is where the phase speed error of the scheme is at its worst. The designationof \wl4" is given to this fourth-order specialization of the general wl scheme.Figure 3.21 shows the scheme's error for this range so it can be compared withother choices of w. For low �, the scheme is accurate to 1% for resolution of N>�6or greater. Table 3.14 gives the error at N = 4 and the required N for �p = 1% and�p = 1360. This is the error of scheme ul1dAd-O4 for the lower half of that scheme'sstability range. These errors will be compared with schemes using other values forw. ��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 r2.98 r1.94 r1.12r0 5.264.744.122 7.256.575.742Scheme wl4 (3.24,3.12) �max = 12 (stable to � = 1)Table 3.14: Error ranges for scheme wl4.

81w = w4th Errors

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

Amplitude

nu

theta

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

Phase

nu

theta

Amplitude

0.10.2

0.30.4

nu

1

2

3

theta

-0.1

-0.075

-0.05

-0.025

0

0.10.2

0.30.4

nu

1

2

3

theta

Phase

0.10.2

0.30.4

nu

1

2

3

theta

0

0.1

0.2

0.3

0.10.2

0.30.4

nu

1

2

3

theta

Scheme wl4 (3.24,3.12) �max = 12 (stable to � = 1)Figure 3.21: Error plots for scheme wl4.

823.7.2 Fifth-Order Scheme: w = �12 + 15� + �5When w = w5(�) = �12 + 15� + �5 (3.25)the fourth-order error term drops from equation (3.24), and the scheme becomes�fth-order accurate. This weight (and others to be discussed below) is shown inFigure 3.22. This scheme is designated \wl5", the �fth-order accurate version. Theerrors for this case are presented in Figure 3.23, and Table 3.15. In contrast toall previous schemes, including the sixth-order accurate schemes, this �fth orderscheme has very accurate phase speed, incurring 1% error at the coarse resolution ofN = 3 and no more than 3% error for the maximum resolvable frequency, N = 2.Compared with the w = 0 case, and the sixth-order upwind leapfrog scheme, thisscheme demonstrates the remarkable di�erence a well-placed point in the stencil canmake to the phase speed accuracy. As expected, to obtain this degree of accuracy,the weight w has to increase to balance the scheme as � is lowered. Interestingly,at � = 12 , where scheme sl4 (using w = 0) is exact, this �fth-order accurate schemechooses w 6= 0 and is not exact.This �fth-order scheme, which is highly accurate in �p, is limited by a poorer �aerror of 1% at N = 6, roughly equivalent to the phase speed error of the fourth-orderscheme. If phase speed and amplitude errors are treated equally, this scheme can besaid to be no better than the fourth-order scheme. But it can be improved upon bya di�erent choice of weight.Figure 3.24 shows the very large negative excess ampli�cation of the spuriousmode and reveals that this scheme has a very rapidly damped spurious mode, espe-cially for low frequencies and low Courant numbers. This would prove to be useful

83

-0.5

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

w

nu

w5(x)wA(x)wB(x)wC(x)

Figure 3.22: w(�) for various optimizations of wl schemes.

84w = w5th Errors

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

Amplitude

nu

theta

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

Phase

nu

theta

Amplitude

0.10.2

0.30.4

nu

1

2

3

theta

-0.1

-0.075

-0.05

-0.025

0

0.10.2

0.30.4

nu

1

2

3

theta

Phase

0.10.2

0.30.4

nu

1

2

3

theta

0

0.01

0.02

0.03

0.10.2

0.30.4

nu

1

2

3

theta

Scheme wl5 (3.24,3.25) �max = 12Figure 3.23: Error plots for scheme wl5.

85��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 r0.17r0.117r0.0695r0 2.722.542.332 3.613.363.052��max 100�a(N = 4) N(�a = 1%) N(�a = 1360)1312231 r -3.07r -2.28r-1.49r0 5.124.794.362 6.736.35.732Scheme wl5 (3.24,3.25) �max = 12Table 3.15: Error ranges for scheme wl5.as strong spurious modes can obscure physical modes.3.7.3 Optimizing in wTo see how stability and accuracy vary with the choice of w, the amplitude andphase errors for a range of w and � are plotted in Figure 3.25. The resolution ofN = 4 is used in Figure 3.25, and although this is rather arbitrary, it serves toillustrate the error incurred for this level of resolution. If the scheme is well-behavedwith respect to N , checking a single resolution is a good place to start. (Althoughstability has not been checked for all modes for all w, � and �, it appears frompreliminary experiences with plots such as Figure 3.24 that the spurious modes areconsistently damped faster than the physical modes and are therefore stable if the

86w = w5th Spurious Mode Excess Amplification

Amplitude

0.10.2

0.30.4

0.5

nu

1

2

3

theta

-1

-0.75

-0.5

-0.25

0

0.10.2

0.30.4

0.5

nu

1

2

3

theta

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

Amplitude

nu

theta

Figure 3.24: Excess ampli�cation of spurious mode of scheme wl5.physical mode is stable. The stability of all modes can be checked a posteriori.)Figure 3.25 shows that a slightly unstable region exists for low (but not zero) w and�. This region must be avoided in choosing w.The contours of �a = �0:01 and �p = +0:01 from Figure 3.25 are isolated inFigure 3.26. They outline a region in which �a and �p are closest to each other inmagnitude. For �>�0:223, it is possible to �nd a weight that gives better than 1%error in both measures, but unfortunately for �<�0:223, as one error decreases, theother increases, such that it is impossible to simultaneously achieve 1% accuracy inboth amplitude and phase. The best that can be done is to choose weight functionsthat lie close to this region. The three curves labeled wA, wB and wC in Figure3.22 are these optimized weights. The schemes using these weights are designated\wlA", \wlB" and \wlC" respectively. (Note that these weights are intended onlyfor 0 � � < 12 . For � > 12, the w = 0 scheme is quite su�cient.) They all have the

87Errors

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4Amplitude

nu

w

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4Phase

nu

w

Amplitude

0.10.2

0.30.4

nu

01

23

4

w

-0.2

-0.1

0

0.10.2

0.30.4

nu

01

23

4

w

Phase

0.10.2

0.30.4

nu

01

23

4

w

-0.025

0

0.025

0.05

0.10.2

0.30.4

nu

01

23

4

w

Figure 3.25: Amplitude and phase errors of wl scheme at N = 4 and variable � andw.

880.1 0.2 0.3 0.4 0.5

0

0.5

1

1.5

2

2.5

3

3.5

4Low Error Region

nu

w

Figure 3.26: Region of small amplitude and phase errors for wl scheme at N = 4.three-parameter form of w5 w(�) = a0 + �a1 + a2� (3.26)and are constrained to pass through two common points in the �-w plane. The �rstis (0:223; 0:23) where ��a � �p � 0:01, and the second is (12 ; 0) since the w = w4 = 0scheme is known to be exact at � = 12 . The last constraint is that wA passes through(0.0975,0.8), which lies on the ��a = 0:01, curve; wB passes through (0.1225,0.8),which lies on the �p = 0:01 curve; and wC passes through (0.1100,0.8) which liesbetween them. Thus, wA gives better amplitude error, wB gives better phase speederror and wC is a compromise4. For 0:223<�� � 12 , both errors should be less than4Numerical values for the weights arewA = �0:2349+ 0:1002=�+ 0:06874� (3.27)wB = �0:7207 + 0:1752=�+ 0:7407� (3.28)wC = �0:4442+ 0:1325=�+ 0:3582� (3.29)

891%, and for �<�0:223, they should ideally be equal, though they cannot both be lessthan 1%.Numerical experiments show little di�erence between the performances of wA,wB and wC, so we concentrate on the compromise, wC. The error for this weight isshown in Figure 3.27. Figure 3.28 shows that the spurious mode is damped fasterthan the physical mode.Numerical values for �p and �a for the wC scheme is given in Table 3.16 for N = 4.The scheme is a successful compromise between using w4 and using w5 in that �pand �a are very close to each other. The worst error between �p and �a is lowerfor the optimized scheme than for the fourth- or �fth-order schemes and in fact,j�pj + j�aj for wC is also less than or very close to max(j�pj; j�aj) for w4 and w5 forall the values shown. Even though a �nite amplitude error is incurred over the errorof the reversible w = 0 scheme, the better overall accuracy of the w = wC schemedemonstrates that optimizing can be feasible and advantageous.3.8 The Next StepIn this chapter, we have dealt exclusively with 1D problems, comparing sev-eral di�usive and non-di�usive schemes and central-di�erence and upstream-biasedschemes. Each type has its strong points. The upstream-biased schemes overallshow better phase speed error than the central-di�erence schemes, or at least, eachperforms better than the other in di�erent regions of the stability ranges. The tworeversible leapfrog types of schemes have no amplitude error, and without having tobe concerned with amplitude errors, we sought to reduce their phase speed errors buusing high-order accurate extensions. But the irreversible schemes which have �nite

90w = wCth Errors

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

Amplitude

nu

theta

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

Phase

nu

theta

Amplitude

0.10.2

0.30.4

nu

1

2

3

theta

-0.1

-0.075

-0.05

-0.025

0

0.10.2

0.30.4

nu

1

2

3

theta

Phase

0.10.2

0.30.4

nu

1

2

3

theta

0

0.05

0.1

0.15

0.10.2

0.30.4

nu

1

2

3

theta

Scheme wlC (3.24,3.29) �max = 12Figure 3.27: Error plots for scheme wlC.

91w = wCth Spurious Mode Excess Amplification

Amplitude

0.10.2

0.30.4

0.5

nu

1

2

3

theta

-1

-0.75

-0.5

-0.25

0

0.10.2

0.30.4

0.5

nu

1

2

3

theta

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

Amplitude

nu

theta

Figure 3.28: Excess ampli�cation of spurious mode of scheme wlC.amplitude errors also have some advantages over the reversible ones. They havemore exible design because they do not have to possess the symmetry imposed onreversible schemes. Although they su�er from amplitude error in addition to phasespeed error, over all, it may be possible to make both errors smaller than the phasespeed error of the reversible schemes. In one case, we used the extra latitude to de-sign an scheme that can be optimized for better phase replication, better amplitudereplication or equal phase and amplitude errors.The new upwind leapfrog schemes performed quite well against the others, jus-tifying further investigation using multi-dimensional problems. They are unique inthat they are the �rst to be both upstream-biased and non-di�usive. Based on thepromising results in 1D and the material presented in Chapter II, particularly inthe use of bicharacteristic equations, the reversible schemes are next extended tosolve multi-dimensional problems and compared to the multi-dimensional standardleapfrog schemes.

92��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 r1.19r0.923r0.623r0 4.193.923.522 5.835.474.952��max 100�a(N = 4) N(�a = 1%) N(�a = 1360)1312231 r-1.22r-0.881r-0.544r0 4.153.93.542 5.2954.582Scheme wlC (3.24,3.29) �max = 12Table 3.16: Error ranges for scheme wlC.

CHAPTER IVSECOND-ORDER STANDARD AND UPWINDLEAPFROG SCHEMES IN TWO AND THREEDIMENSIONSIn this Chapter, the second-order standard leapfrog and upwind leapfrog schemesin 2D and 3D are described, analyzed and compared for both the acoustics equationsand the electromagnetics equations.The descriptions for the bicharacteristic schemes will be more detailed, since theyare a development of this work. The stability limits and numerical errors of the twoapproaches are discussed and compared.4.1 Two-Dimensional Acoustics and ElectromagneticsBecause the 2D acoustic equations can be transformed into both the 2D TE and2D TM electromagnetic equations by a simple transformation of variables (Section2.3), the scheme behaves identically for these problems in 2D, in exact and numericalformulations. Therefore, only the acoustics equations are treated here. The resultsapply also to the 2D TE and 2D TM equations.93

944.1.1 Standard leapfrogIn 2D, the standard leapfrog scheme uses the stencil in Figure 4.1. Pressure isstored at the cell center and normal velocity components are on the faces. Pressureand velocity are stored at alternating time steps, as depicted in the �gure. Thisscheme (and its high-order counterparts presented later) requires the storage of three oating-point numbers per cell for the solution. Thus resolving a 2D domain of size` � ` with resolution N requires 3N2 oating-point numbers for the solution. Thismethod uses central-di�erence approximations for equations (2.14) to obtainpn+1;i;j � pn+1;i;j2�� + c ��xn;i;ju +�yn;i;jv� = 0 (4.1a)un+1;i;j � un+1;i;j2�� + c�xn;i;jp = 0 (4.1b)vn+1;i;j � vn+1;i;j2�� + c�yn;i;jp = 0 (4.1c)where � is the second order accurate central-di�erence operators described in Ap-pendix D. Note that the equations are applied to the location of the variable beingupdated, not at the same point for all variables. This formulation is designated bythe name \sl2dAc-O2" (the third pair of letters stand for the acoustic problem).It results in a system with six ampli�cation factors, two of which are stationary(� = �1). The moving mode ampli�cation factors are8>>>>>>>>>>><>>>>>>>>>>>: qa+pa2 � 1�qa+pa2 � 1qa�pa2 � 1�qa�pa2 � 1 9>>>>>>>>>>>=>>>>>>>>>>>; (4.2)where a = 1 � 4�2(2 � cos �x � cos �y)

95xy�vv nu n� 1

n+ 1ppu(a) pressure update stencil nu n� 1

n+ 1p pu(b) u-velocity update stencilFigure 4.1: Standard leapfrog stencil for two-dimensional acoustics. Bold lines aregrid lines. Fine lines show where �nite-di�erences are made. Presureis stored at cell centers. Normal velocity components are stored at facecenters.The factors are for the positive real mode (mode moving in the direction �xbx+ �yby),the negative real mode (mode moving in the direction ��xbx��yby), and two spuriousmodes, respectively. While these modes are stable for 0 � � � 12 when �2 = �2x+�2y ��2 (the 1D maximum phase angle), they are stable for all 0 � �x; �y � � only when0 � � � �max = 12p2 � 0:35355. The errors for this scheme are presented below andcompared with that of the second-order upwind leapfrog scheme for equations (2.14).4.1.2 Upwind leapfrogIn the 1D upwind leapfrog scheme, the dependent variables are located at thegrid points, and the update stencil is centered about the midpoint between two con-secutive points. In higher dimensions, the appropriate generalization of the midpointis a cell centroid and the appropriate generalization of the points are the faces on thesurface of the cell. This leads to a scheme in which variables (pressure and normalcomponent of velocity or G�n) are stored on the face centroids and the di�erencestencil is centered on the cell centroid as depicted by Figure 4.2. Computer codes

96xy�G�xG�xG+y n� 1G+yG+y

G+y nn+ 1(a) G+y update stencil

G�yG�x G�x n � 1G�x G�y G�x nn + 1(b) G�x update stencilFigure 4.2: Upwind leapfrog stencil for 2D bicharacteristic equations. Thick lines aregrid lines. Thin lines are lines along which �nite-di�erences are taken.Characteristic variables in the directions normal to a face are stored onthe face.for upwind leapfrog schemes can be written with the newest solution overwriting theoldest solution, so that only two time levels of solution are needed. This scheme (andits higher-order counterparts) requires the storage of eight oating-point numbers percell for the 2D scheme. Thus resolving a 2D domain of size ` � ` with resolution Nrequires 8N2 oating-point numbers for the solution.The 1D scheme (Figure 3.7 and equation (3.8)) is extended by including a cell-centered di�erence for the cross term. The x-bicharacteristic equation is extended bya di�erence along the y-direction, and vice versa. Figure 4.2 shows only the stencilfor the G�x and G+y waves. The stencil for the G+x and G�y waves are mirrorimages of the ones shown. Applying this scheme to the 2D acoustics equations givesthe discrete equationsG+xn+1;i+1=2;j �G+xn;i+1=2;j +G+xn;i�1=2;j �G+xn�1;i�1=2;j2��+c�xn;i;jG+x + c2�yn;i;j �G+y �G�y� = 0 (4.3a)

97G�xn+1;i�1=2;j �G�xn;i�1=2;j +G�xn;i+1=2;j �G�xn�1;i+1=2;j2���c�xn;i;jG�x + c2�yn;i;j �G+y �G�y� = 0 (4.3b)G+yn+1;i;j+1=2 �G+yn;i;j+1=2 +G+yn;i;j�1=2 �G+yn�1;i;j�1=22��+c�yn;i;jG+y + c2�xn;i;j �G+x �G�x� = 0 (4.3c)G�yn+1;i;j�1=2 �G�yn;i;j�1=2 +G�yn;i;j+1=2 �G�yn�1;i;j+1=22���c�yn;i;jG�y + c2�xn;i;j �G+x �G�x� = 0 (4.3d)These equations apply at each cell center, indexed (i; j) on the spatial grid, althoughno variable exists there. All the variables are displaced a half grid index from thecoordinate (i; j). (By applying these equations at one grid cell at a time, one canupdate all the face-centered variables without having to make redundant computa-tions on the cross terms. Hence it is called a face-centered storage, cell-centeredupdate scheme.) This formulation is designated by the name \ul2dAc-O2". It hasan ampli�cation matrix with the characteristic equation�8 + a71�7 + a62�6 + a53�5 + a44�4 + a53�3 + a62�2 + a71� + 1 = 0 (4.4)where a71 = �(�4a1 + 8) + (2a1 � 4)a62 = �2(�20a1 + 6a2 + 28) + �(24a1 � 8a2 � 32) + (�8a1 + 2a2 + 8)a53 = �3(�32a1 + 16a2 + 32) + �2(80a1 � 40a2 � 80)��(�60a1 + 32a2 + 56) + (14a1 � 8a2 � 12)a44 = �3(64a1 � 32a2 � 64) + �2(�120a1 + 68a2 + 104)+

98�(80a1 � 48a2 � 64) + (�16a1 + 12a2 + 14)a1 = cos �x + cos �ya2 = cos(�x + �y) + cos(�x � �y)for the ampli�cation factor �, Even for such a simple scheme (second-order accurate,2D), the von Neumann analysis has become very di�cult to perform symbolically.All von Neumann analyses from this point on use the numerical procedure outlinedin Appendix C. That analysis shows that the stability limit of the ul2dAc-O2 schemeis �max = 12 .4.1.3 Numerical Errors and ComparisonsIn two dimensions, the numerical errors are, in addition, dependent on the prop-agation angle � = tan�1 �y�x! (4.5)The errors for � = �max and � = 12�max, � 2 h0; �4 i and a range of � up to � areshown in Figure 4.3 for the sl2dAc-O2 scheme, and Figure 4.4 for the ul2dAc-O2scheme. The �gures show that at � = �max for the respective schemes, the ul2dAc-O2 scheme is exact along the axes and worst at � = �4 , while the opposite is true forthe sl2dAc-O2 scheme.The ranges of �p over all propagation angles for N = 4 are summarized in Tables4.1 and 4.2, as are the resolutions required to bring the worst �p to one percent or 1360.These errors and those of the schemes to follow are very well behaved in N , so thatthe errors for higher values of N can be approximately deduced from the error atN = 4 and the order of the truncation error of the scheme. For N = 4 at the stabilitylimit of each individual scheme, the sl2dAc-O2 scheme has �p 2 [�5:7%; 0] while theul2dAc-O2 scheme has �p 2 [�2:7%; 0] over all �. Both schemes err on the slow side

9945

90135

180

015

3045

-0.3-0.2-0.1

0

� �� = �max� = 12�max

Figure 4.3: Phase error plots for scheme sl2dAc-O2.45

90135

180

015

3045

-0.10

0.10.20.3

� �� = �max� = 12�max

Figure 4.4: Phase error plots for scheme ul2dAc-O2.

100��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !-9.55 -4.57!-9 -3.93!-8.21 -2.99!-5.72 0 12.51211.39.16 23.622.821.517.3sl2dAc-O2 (4.1) �max = 12p2Table 4.1: Error ranges for scheme sl2dAc-O2.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !3.74 12.5!1.21 7.98!-0.64 4.56!-2.71 0 13.611.18.556.48 25.721.116.212.2ul2dAc-O2 (4.3) �max = 12Table 4.2: Error ranges for scheme ul2dAc-O2.at � = �max, but as � is decreased, the standard leapfrog scheme drops further, whilethe upwind leapfrog scheme speeds up. For a while, this means that the upwindleapfrog scheme's error is being improved, but it soon overshoots and becomes toofast. To attain �p = 1% at � = �max, scheme sl2dAc-O2 requires N = 9:16 andscheme ul2dAc-O2 requires N = 6:48The discussion on the actual costs (memory and oating-point operations re-quired) will be deferred until all schemes have been studied. This discussion ispresented in Chapter VI.Because both schemes are exact for certain angles of propagation, the maximum

101error is also the anisotropic dispersion error (for � = �max). Away from � = �max,the di�erence between the fastest error and the slowest error for a �xed � is a directmeasure of anisotropy error. This is indicated in the tables by the bar drawn fromthe slowest error to the fast. Anisotropy error is important in applications wherewaves from di�erent directions converge at the same point. The anisotropy error isclearly evident in the Figures 4.3 and 4.4, for any �-constant cut. At their stabilitylimits, the upwind leapfrog scheme has signi�cantly better anisotropy error than thestandard leapfrog scheme. But as �p degrades, so does the corresponding anisotropyerror, and at around � = 23�max, it is worse for upwind leapfrog than for standardleapfrog. Somewhere between � = 12�max and � = 13�max, the �p itself becomes worsefor upwind leapfrog than for standard leapfrog. This pattern of degrading error, withone scheme outperforming the other in di�erent regions of the stability range, is verysimilar to that observed in 1D.At � = 23�max, the worst errors are -8.2% and -4.6% respectively, and neitherscheme is exact at any �. The signi�cance of this Courant number is that it isthe maximum usable Courant number for many problems in many electromagneticscattering problems involving material with index of refraction close to 1.5, such asglass1.4.2 Three-Dimensional AcousticsIn three dimensions, the divergence-driven acoustics equations and the curl-drivenMaxwell's equations require di�erent staggering strategies. This Section covers theschemes for the acoustics equations, and the next covers the schemes for the electro-1For problems with two materials, A and B, whose speeds of propagation satisfy cA � 1:5cB(for example, vacuum and glass), the maximum time step possible corresponds with �A = �max and�B � 23�max, regardless of the value of �max. The data for � = 13�max apply when the index ratio isthree.

102magnetics equations.4.2.1 Standard leapfrogFigure 4.5 shows the storage arrangements required by the standard leapfrogscheme for acoustics. This scheme (and its higher-order counterparts) requires stor-age for four oating-point numbers per cell for the solution. Pressure is stored atthe cell centers at every other time step, and normal velocity components are storedat the face centers at the remaining time steps, analogous to the 2D situation. Thusresolving a 3D domain of size `� `� ` with resolution N requires 4N3 oating-pointnumbers for the solution. The space-times stencil used are direct extensions of thosein 2D (Figure 4.1). Applying this scheme to the 3D acoustics equations 2.13 givesthe di�erence equationspn+1;i;j;k � pn;i;j;k2�� + c ��xn;i;j;ku +�yn;i;j;kv +�zn;i;j;kw� = 0 (4.6a)un+1;i;j;k � un;i;j;k2�� + c�xn;i;j;kp = 0 (4.6b)for pressure and x-component of velocity. The remaining equations are similar. Thisscheme is designated \sl3dAc-O2". The stability limit for this scheme is �max = 12p3.The trend appear to be that in k-dimensions, the stability limit of the standardleapfrog scheme is �max = 12pk .4.2.2 Upwind leapfrogFigure 4.6 shows the arrangements required by the upwind leapfrog scheme. Thescalar direction-associated characteristic variables G�n or alternatively, pressure andthe component of velocity along bn, are stored at the centers of faces normal to thelongitudinal directions bn. Nothing is stored at the cell center. Again, the newesttime level can overwrite the oldest so that only two time levels need to be stored.

103xyzp uu w

wv vFigure 4.5: Standard leapfrog storage scheme for 3D acoustics equations. Thick linesare grid lines. Thin lines are lines along which �nite di�erences are made.Pressure is stored at cell center. Normal velocity components are storedon faces.This scheme (and its higher-order counterparts) stores 12 oating-point numbers percell for the solution, Thus resolving a 3D domain of size `� `� ` with resolution Nrequires 12N3 oating-point numbers for the solution. The space-time stencils usedare direct extensions of those in 2D (Figure 4.2) but with more cross terms.Applying this scheme to the equations for G�x givesG+xn+1;i+1=2;j;k �G+xn;i+1=2;j;k +G+xn;i�1=2;j;k �G+xn�1;i�1=2;j;k2��+c�xn;i;j;kG+x + c2 h�yn;i;j;k �G+y �G�y�+�zn;i;j;k �G+z �G�z�i = 0 (4.7a)G�xn+1;i�1=2;j;k �G�xn;i�1=2;j;k +G�xn;i+1=2;j;k �G�xn�1;i+1=2;j;k2���c�xn;i;j;kG�x + c2 h�yn;i;j;k �G+y �G�y�+�zn;i;j;k �G+z �G�z�i = 0 (4.7b)The equations for G�y and G�z are similar. This scheme is designated \ul3dAc-O2".The stability limit is �max = 13 , which is, as in the 2D case, greater than that ofthe standard leapfrog version. Like the standard leapfrog scheme, the stability limit

104xyzG�z G�xG�yG�zG�x G�yFigure 4.6: Upwind leapfrog storage scheme for 3D acoustic bicharacteristic equa-tions. Thick lines are grid lines. Thin lines are lines along which �nitedi�erences are made. Scalar characteristic variables associated with theface-normal direction are stored on each face.of this scheme appears to be decreasing with the number of spatial dimensions. Ink-dimensions, the stability limit of this scheme seems to be �max = 1k . The stabilitylimit is higher for the upwind leapfrog scheme, for k � 4, which includes all practicalreal problems.4.2.3 Numerical Errors and ComparisonsIn 3D, as in 2D, the phase speed depends on the direction of propagation. Thedirection in 3D is parameterized by the angles � and � shown in Figure 4.7, where� is the rotation angle from the z-axis toward the x-axis and � is the rotation anglefrom the z-axis toward the y-axis. Note that (� = 0; � = 0) corresponds to acoordinate direction, while (� = �4 ; � = �4 ) corresponds to the angle most skewedfrom all coordinate directions in 3D. Due to symmetry, only data for � 2 h0; �4 i and� 2 h0; �4 i are presented. And in fact, only half of that is really necessary, since

105z x�� yFigure 4.7: De�nition of parametric angles in 3D.switching � and � yields a symmetric direction2.Both schemes have very well behaved errors with respect to the grid resolution,so we present the error plots for only a resolution of N = 4. The error plots of thestandard leapfrog scheme are shown in Figure 4.8, and the error plots of the upwindleapfrog scheme are shown in Figure 4.9 for � = �max and � = 12�max. Figure 4.8shows that at � = �max, the standard leapfrog scheme is exact at the most skewedangle, which was the case in 2D, except that in 2D, �max and the most skewedan are di�erent. The same trend from 2D does not carry to 3D for the upwindleapfrog scheme because the upwind leapfrog scheme is exact for the 1D and 2Daxial directions at � = 12 , but in 3D, � = 12 is not within the stable range. Figure 4.9clearly shows the upwind leapfrog scheme to be fast at (� = 0; � = 0). Nevertheless,the scheme is still exact at one point in the �� � space shown and at 24 distinct but2These parameters are chosen because for �; � 2 �0; �4 �, the spherical surface can be mappedto a at square without much distortion. This removes much of the optical area-bias towardcertain regions on the spherical surface when comparing the standard leapfrog scheme to the upwindleapfrog scheme. As shown below, each scheme performs its best in di�erent directions. (Thiscoordinate system also makes for a more e�cient coverage of the angles in numerical eigenvaluecomputations described in Appendix C). The region close to (� = �4 ; � = �4 ) still appear slightlyenlarged compared to the region close to (� = 0; � = 0), but not to the extent of that usingconventional spherical coordinates.

106symmetrically similar directions in 3D.0 15 30 45

015

3045

-0.08-0.06-0.04-0.02

0

� �� = �max� = 12�max

Figure 4.8: Phase error plots for scheme sl3dAc-O2.0 15 30 45

015

3045

00.050.1

� �� = �max� = 12�max

Figure 4.9: Phase error plots for scheme ul3dAc-O2.Numerical values of �p for are given in Tables 4.3 and 4.4. The tables showthat at � = �max, the error of the upwind leapfrog scheme is about one-third less

107than that of the standard leapfrog scheme. At � = 23�max, they are very roughlyequal, with the standard leapfrog scheme slightly better. And at � = 13�max, theerror of the standard leapfrog scheme is very roughly one-third less than that of theupwind leapfrog scheme. To attain �p = 1% at � = �max, scheme sl3dAc-O2 requiresN = 10:5 and scheme ul3dAc-O2 requires N = 8:55 The corresponding cost for theseschemes are discussed in Chapter VI.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !-9.69 -3.05!-9.33 -2.6!-8.82 -1.96!-7.26 0 12.612.311.810.5 23.923.322.519.9sl3dAc-O2 (4.6) �max = 12p3Table 4.3: Error ranges for scheme sl3dAc-O2.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !2.81 16.3!1.09 12.5!-0.313 9.34!-2.33 4.56 15.213.6128.55 28.625.722.616.2ul3dAc-O2 (4.7) �max = 13Table 4.4: Error ranges for scheme ul3dAc-O2.Compared with the 2D system and the 3D electromagnetic system presented inthe next section, this is the worst performance of the second-order upwind leapfrog

108scheme relative to the second-order standard leapfrog scheme. The problem may bebecause the stability limit for the upwind leapfrog scheme here is signi�cantly lower(�max = 13 rather than 12), and as we have seen, error is higher for lower �. We shallsee in the next section that the upwind leapfrog scheme for electromagnetics has astability limit that (perhaps surprisingly) does not degrade from 2D and that schemehas better errors for the same resolution. The stencils and discretizations for both theacoustic and the electromagnetic systems of equations are virtually the same usingthe upwind leapfrog scheme. The fact that when applied to the acoustic system ofequations, the upwind leapfrog scheme's stability limit drops is at the present timea mystery. (But there is yet more mystery in the high-order schemes.)4.3 Three-Dimensional Electromagnetics4.3.1 Standard leapfrogFigure 4.10 shows the storage arrangement required by the standard leapfrogscheme for electromagnetics. Components of H normal to each face are kept withthe face. Components of E along each grid line are kept with the line. Nothing iskept at the cell centers. This scheme is identical to Yee's scheme [14] as shown inAppendix B. It (and its higher-order counterparts) requires storing six oating-pointnumbers per cell for the solution. Thus resolving a 3D domain of size `� `� ` withresolution N requires 6N3 oating-point numbers for the solution.Relative to the point being updated, the di�erence scheme is a simple centraldi�erence scheme. The di�erence equation for Ez isEzn+1;i;j;k � Ezn+1;i;j;k2�� + c ��yn;i;j;kHx ��xn;i;j;kHy� = 0 (4.8)where (i; j; k) are the grid indices of the point being updated. The equations for theother variables are similar. This scheme is designated \sl3dEm-O2", where the third

109xyz

HzEz ExEx HxHyEy EyEyEyEz EzEzEx ExHzHyHx

Figure 4.10: Standard leapfrogstorage scheme for 3D electromagnetics equations.Thick lines are grid lines. Thin lines are lines on which �nite di�erencesare made. Normal H components are stored on faces. Longitudinal Ecomponents are stored at line midpoints.pair of letters stands for the electromagnetics problem.The stability limit of the sl3dEm-O2 scheme is �max = 12p3 , the same as for the3D acoustics equations (and they follow the same trend).4.3.2 Upwind leapfrogFigure 4.11 show the arrangements required by the upwind leapfrog scheme forelectromagnetics. The two transverse components of the characteristic variable vec-tors (or alternatively the transverse components of the �eld vectors) are kept at theface centers. Longitudinal components are not stored. Once again, storage of onlytwo levels of solutions is needed. This scheme requires storing 24 oating-point num-bers per cell for the solution space. Thus resolving a 3D domain of size `� `� ` withresolution N requires 24N3 oating-point numbers for the solution. The space-timestencil is an extension of the 2D scheme (Figure 4.2) and is similar to the 3D schemefor acoustics except that there are two waves for each of the six directions, because

110there are two polarizations. In addition, there are two cross terms for each equationrather than one. Geometrically, the di�erence stencils are the same for the acousticequations and the electromagnetic equations.The di�erence equations for the characteristic variables along the x-direction areGxyn+1;i+1=2;j;k �Gxyn;i+1=2;j;k +Gxyn;i�1=2;j;k �Gxyn�1;i�1=2;j;k2�� +ci;j;k�xn;i;j;kGxy�ci;j;k2 h�yn;i;j;k(Gyxy + Syxy ) + �zn;i;j;k(�Gzyz + Szyz )i = 0 (4.9a)Sxyn+1;i�1=2;j;k � Sxyn;i�1=2;j;k + Sxyn;i+1=2;j;k � Sxyn�1;i+1=2;j;k2�� �ci;j;k�xn;i;j;kSxy�ci;j;k2 h��yn;i;j;k(Gyxy + Syxy ) + �zn;i;j;k(�Gzyz + Szyz )i = 0 (4.9b)Gxzn+1;i+1=2;j;k �Gxzn;i+1=2;j;k +Gxzn;i�1=2;j;k �Gxzn�1;i�1=2;j;k2�� +ci;j;k�xn;i;j;kGxz�ci;j;k2 h�zn;i;j;k(Gzxz + Szxz )��yn;i;j;k(Gyzy � Syzy )i = 0 (4.9c)Sxzn+1;i�1=2;j;k � Sxzn;i�1=2;j;k + Sxzn;i+1=2;j;k � Sxzn�1;i+1=2;j;k2�� �ci;j;k�xn;i;j;kSxz�ci;j;k2 h��zn;i;j;k(Gzxz + Szxz )��yn;i;j;k(Gyzy � Syzy )i = 0 (4.9d)This scheme is designated \ul3dEm-O2". As noted previously, for clarity, the vari-able S is used to represent the characteristic variable in moving in reverse. A secondidenti�er has been introduced for the characteristic variables as as they are brokendown into their scalar components. The �rst identi�es the direction associated withthe propagation direction, and the second is the polarization direction (of the char-acteristic variable or E). For example, Gxy is the y-component of the characteristicvariable vector moving in the positive x-direction, and Sxy is the y-component of thecharacteristic variable vector moving in the negative x-direction. Many terms appear

111xyzG�yG�z G�xG�x G�y G�z

Figure 4.11: Upwind leapfrog storage scheme for 3D electromagnetic bicharacteristicequations. Thick lines are grid lines. Thin lines are lines on which �nitedi�erences are mdae. Vector characteristic variables associated withnormal direction are stored on each face (similar to the stencil for theacoustic scheme ul3dAc-O2).in more than one equations, and redundant computations can be avoided by using acell-centered update procedure.The stability limit of the ul3dEm-O2 scheme is �max = 12 , the same as it is in2D, which is surprising because all the previous schemes, both standard leapfrog andupwind leapfrog have reduced stability limits when extended from 1D to 2D andfrom 2D to 3D.4.3.3 Numerical Errors and ComparisonsThe errors for sl3dEm-O2 and ul3dEm-O2 are plotted in Figures 4.12 and 4.13 re-spectively. Like all previous multi-dimensional cases, except the 3D upwind leapfrogscheme for acoustics, the standard leapfrog scheme is best at the most skewed angleand the upwind leapfrog scheme is best along the axes. Each scheme is exact at itsstability limit and along its respective best directions.

1120 15 30 45

015

3045

-0.08-0.06-0.04-0.02

0

� �� = �max� = 12�max

Figure 4.12: Phase error plots for scheme sl3dEm-O2.0 15 30 45

015

3045

0

0.05

� �� = �max� = 12�max

Figure 4.13: Phase error plots for scheme ul3dEm-O2.

113Tables 4.5 and 4.6 show some numerical values of �p for these schemes. For� = 23�max and � = �max, the worst error for the upwind leapfrog scheme is roughlyhalf that of the standard leapfrog scheme. At � = 13�max, they are roughly equal,with the standard leapfrog scheme slightly more accurate. This behavior is similarto the 2D problem. To attain �p = 1% at � = �max, scheme sl3dEm-O2 requiresN = 10:5 and scheme ul3dEm-O2 requires N = 7:65 The corresponding cost forthese schemes are discussed in Chapter VI.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !-9.69 -3.05!-9.33 -2.6!-8.82 -1.96!-7.26 0 12.612.311.810.5 23.923.322.519.9sl3dEm-O2 (4.8) �max = 12p3Table 4.5: Error ranges for scheme sl3dEm-O2.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !1.02 12.5!-1.07 7.98!-2.62 4.56!-4.43 0 13.611.18.557.65 25.721.116.214.2ul3dEm-O2 (4.9) �max = 12Table 4.6: Error ranges for scheme ul3dEm-O2.Interestingly, the error of the standard leapfrog scheme for electromagnetics isidentical to that for acoustics when the analysis parameters (�, �, � and N) are the

114same. But this is not the case for the upwind leapfrog scheme. Figure 4.14 showsthat �p;Ac � �p;Em is positive for N = 4, � = 13 and � = 16 for every propagationangle except those corresponding to 2D and 1D problems. The explanation of whythe speeds are the same for the standard leapfrog scheme but not for the upwindleapfrog scheme has not been found. The di�erences in propagation speeds in any ofthe 2D directions are clearly zero, as expected.0 15 30 45

015

3045

0.0005

0.0015

0.0025

� �� = 13�max� = 16�max

Figure 4.14: Di�erence in �p between schemes ul3dAc-O2 and ul3dEm-O2.4.4 Other ConsiderationsIn addition to the error presented here and the cost to be presented in ChapterVI, there are other issues that may be considered.It is interesting to contrast the upwind leapfrog and standard leapfrog schemeson the scale of convenience and ease of use. The upwind leapfrog scheme uses highlycompact stencils that do not overlap and do not span more than one cell. Some ofthis compactness is lost in the higher-order versions described in Chapter V, but the

115the upwind leapfrog schemes are always more compact than the standard leapfrogschemes of the same order. This is desirable at boundaries, whether computationalboundaries or domain partition boundaries in parallel computer implementations.The upwind leapfrog scheme for 3D electromagnetics was be very neatly, elegantlyand e�ciently implemented on massively parallel message-passing computers in refer-ence [3]. The co-location of �eld variables on grid boundaries in the upwind leapfrogschemes are very convenient for boundaries and for computing surface integrals ofquantities relevant to the problems solved. It is also less limiting for more com-plex modeling as dependent quantities interact at certain points such as impedanceboundary conditions. In some respect, the upwind leapfrog schemes present severalconveniences not a�orded by the more economical standard leapfrog schemes.The upwind leapfrog scheme also only uses cell-centered stencils rather than,whereas the standard leapfrog schemes' stencils that are distributed over face centers,cell centers and line centers. When used with an in-homogeneous medium, materialproperties need only be stored at the cell centers for the upwind leapfrog schemes.The standard leapfrog schemes must either store material properties at more placeson the grid cell or compute from near-by locations them on the y. In either case,the e�ciency of the standard leapfrog schemes is reduced. In addition to this, theupwind leapfrog scheme is able to use coarser resolutions, resulting in fewer cells.These numbers cannot be quanti�ed for all cases, since they vary from problem toproblem. It is nevertheless important to remember that the cost of storing andcomputing solutions are not the only costs for a simulation.

CHAPTER VHIGH-ORDER LEAPFROG SCHEMES IN TWOAND THREE DIMENSIONSThis chapter examines high-order schemes that are extensions of the second-orderaccurate multi-dimensional schemes of Chapter IV. The motivation behind this is toattain greater accuracy for the same grid resolution and/or to improve convergencewhen resolution is increased. Not surprisingly, the high-order schemes presentedin this chapter are consistently more accurate than their lower-order counterparts.These high-order schemes use the same grid and storage layout of the second-orderversions, so their memory requirements for a given resolution are the same. Theyare more complex and require more computational operations to update a cell, butthey more than make up for this by requiring coarser resolutions for the same levelof accuracy. A di�culty in using them is that near boundaries, their extendedstencils require more points that are not in the grid. This di�culty is not limited tothe schemes in this work but is prevalent in all numerical discretizations of partialdi�erential equations. It is not a problem with any of the second-order upstream-biased schemes because their cell centered stencils lie completely inside the grid. Thehigh-order upstream-biased schemes do not fully have this advantage, but in general,the upstream-biased schemes tend to keep more of their stencils inside the grid, thus116

117improving the situation. For the present time, we will overlook this di�culty as othere�orts continue to address it.In this chapter, the basic strategy for deriving a higher-order scheme from alower-order scheme is to write the equivalent di�erential equations correspondingto the lower-order di�erence equations. The time-di�erential terms are changed tospace-di�erential terms via Lax-Wendro� substitutions [56] using the set of originalpartial di�erential equations. The space-di�erential terms can be easily resolved onan extended stencil using the �nite di�erence operators described in Appendix D. Aslightly di�erent approach, using compact time-extended stencils rather than space-extended stencils can be found in [50]. Resolving the lowest-order truncation errorterms in an equivalent di�erential equation leaves a higher-order approximation ofthe di�erential equation.5.1 Three-Dimensional AcousticsThis section presents the high-order extensions to the second-order schemes foracoustics. Fourth and sixth order extensions to the standard leapfrog scheme andseveral variations of the fourth-order extension to the upwind leapfrog scheme arepresented.5.1.1 Standard leapfrogThe second-order standard leapfrog scheme for acoustics (sl3dAc-O2) results inthe equivalent di�erential equationpn+1;i;j;k�pn�1;i;j;k2�� + cun;i+1=2;j;k�un;i�1=2;j;kh+cvn;i;j+1=2;k�vn;i;j�1=2;kh + cwn;i;j;k+1=2�wn;i;j;k�1=2h= p� + c(uz + vy + wz) + ch2 (1�4�2)24 nuxxx + vyyy + wzzzo

118�ch2 �26 nuxyy + uxzz + vxxy + vyzz + wxxz + wyyzo+O(h4) (5.1)for pressure and similar equivalent di�erential equations for the velocity components.Resolving the second-order error terms using the stencil in Figure 5.1c (for sl3dAc-O4) produces the fourth-order scheme with the equivalent di�erential equationpn+1;i;j;k�pn�1;i;j;k2�� + cun;i+1=2;j;k�un;i�1=2;j;kh+cvn;i;j+1=2;k�vn;i;j�1=2;kh + cwn;i;j;k+1=2�wn;i;j;k�1=2h�ch2 (1�4�2)24 n�xxxn;i;j;ku +�yyyn;i;j;kv +�zzzn;i;j;kwo+ch2 �26 n�xyyn;i;j;ku +�xzzn;i;j;ku +�xxyn;i;j;kv +�yzzn;i;j;kv +�xxzn;i;j;kw +�yyzn;i;j;kwo= p� + c(uz + vy + wz)� ch4 (1�4�2)(9�4�2)1920 nuxxxxx + vyyyyy + wzzzzzo+ch4 �2(10�6�2)720 nuxyyyy + uxzzzz + vxxxxy + vyzzzz + wxxxxz + wyyyyzo+ch4 �2(�12�2)720 nuxyyzz + vxxyzz + wxxyyzo+ch4 �2(5�12�2)720 nuxxxyy + uxxxzz + vxxyyy + vyyyzz + wxxzzz + wyyzzzo+O(h6) (5.2)A similar procedure is applied to the equations for u, v and w to complete theextension to a fourth-order accurate scheme. The stencil used for u is shown in Fig-ure 5.1d. The corresponding stencils for the second-order standard leapfrog schemefor acoustics is presented in Figures 5.1a and b for comparison.The formulae for the discrete derivatives used are given in Appendix D. We stressagain that these choices for the discretization of the error terms are not unique. Eachis chosen because it is the most compact stencil on which a speci�c error term canbe discretized. No e�ort was made to look at alternative stencils which may resultin lower errors, and this may be a direction for future work.The stability limit for this fourth-order scheme is the same as that for the second-

119sl3dAc-O2 pressure stencil sl3dAc-O2 u-velocity stencilsl3dAc-O4 pressure stencil sl3dAc-O4 u-velocity stencilsl3dAc-O6 pressure stencil sl3dAc-O6 u-velocity stencilStencils to update pressure in cen-ter cell. Requires normal veloc-ity component on all faces depictedplus interior faces. Stencils to update u-velocity oncenter face. Requires pressure in allcells depicted plus interior cells.

xyz

Figure 5.1: Stencils for sl3dAc schemes.

120order scheme, �max = 12p3 . Figure 5.2 shows how �p varies over propagation directionsfor a discretization of N = 4 at � = �max and � = 12�max. Compared with the second-order scheme (Figure 4.8), the errors are smaller by an expected factor of four, butnot much has changed in how the errors are distributed. The worst dispersion erroris still along the axis and the best is at the most skewed angle. And the scheme isstill exact for the most skewed angle at � = �max.0 15 30 45

015

3045

-0.02-0.015-0.01

-0.0050

� �� = �max� = 12�max

Figure 5.2: Phase error plots for scheme sl3dAc-O4.To go one step further in the high-order extension, the error terms in (5.2) canbe resolved to form the sixth-order discretizationpn+1;i;j;k�pn�1;i;j;k2�� + cun;i+1=2;j;k�un;i�1=2;j;kh+cvn;i;j+1=2;k�vn;i;j�1=2;kh + cwn;i;j;k+1=2�wn;i;j;k�1=2h�ch2 (1�4�2)24 n�xxxn;i;j;ku +�yyyn;i;j;kv +�zzzn;i;j;kwo+ch2 �26 n�xyyn;i;j;ku +�xzzn;i;j;ku +�xxyn;i;j;kv +�yzzn;i;j;kv +�xxzn;i;j;kw +�yyzn;i;j;kwo+ch4 (1�4�2)(9�4�2)1920 n�xxxxxn;i;j;ku +�yyyyyn;i;j;kv +�zzzzzn;i;j;kwo

121�ch4 �2(10�6�2)720 n�xyyyyn;i;j;ku +�xzzzzn;i;j;ku +�xxxxyn;i;j;kv +�yzzzzn;i;j;kv +�xxxxzn;i;j;kw +�yyyyzn;i;j;kwo�ch4 �2(�12�2)720 n�xyyzzn;i;j;ku +�xxyzzn;i;j;kv +�xxyyzn;i;j;kwo�ch4 �2(5�12�2)720 n�xxxyyn;i;j;ku +�xxxzzn;i;j;ku +�xxyyyn;i;j;kv +�yyyzzn;i;j;kv +�xxzzzn;i;j;kw +�yyzzzn;i;j;kwo= p� + c(uz + vy + wz) + O(h6) (5.3)for p. The equations for the velocity components follow by a similar procedure. Thesixth-order scheme uses the stencils like those in Figures 5.1e and f.The stability limit remains �max = 12p3 for this scheme. Figure 5.3 shows how �pvaries for discretizations of N = 4 at � = �max and � = 12�max. The worst dispersionerror is along the axis and the best is at the most skewed angle, as before.0 15 30 45

015

3045

-0.006-0.004-0.002

0

� �� = �max� = 12�max

Figure 5.3: Phase error plots for scheme sl3dAc-O6.Tables 5.1 and 5.2 presents �p for these schemes for N = 4 and the resolutionrequired for �p � 1%. The sixth-order scheme attains the goal of one percent errorfor N = 4. Of course, the price for this scheme is that it carries a huge and cumber-some stencil. Like previous standard leapfrog schemes, these schemes degrade fairly

122gracefully for low �. Like the second-order scheme, �p � 0 for all �, resolutions andpropagation angles.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !-2.39 -0.268!-2.29 -0.226!-2.16 -0.167!-1.75 0 5.044.984.894.61 7.016.926.796.38sl3dAc-O4 (5.2) �max = 12p3Table 5.1: Error ranges for scheme sl3dAc-O4.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !-0.753 -0.031!-0.722 -0.026!-0.677 -0.0191!-0.546 0 3.83.773.723.59 4.794.754.694.51sl3dAc-O6 (5.3) �max = 12p3Table 5.2: Error ranges for scheme sl3dAc-O6.5.1.2 Upwind leapfrogThe second-order upwind leapfrog scheme ul3dAc-O2 (4.7) for acoustics yieldsthe equivalent di�erential equationGxn+1;i+1=2;j;k�Gxn;i+1=2;j;k+Gxn;i�1=2;j;k�Gxn�1;i�1=2;j;k2�� + cGxn;i+1=2;j;k�Gxn;i�1=2;j;kh+ c2 Gyn;i;j+1=2;k�Gyn;i;j�1=2;k�Syn;i;j+1=2;k+Syn;i;j�1=2;kh

123+ c2 Gzn;i;j;k+1=2�Gzn;i;j;k�1=2�Szn;i;j;k+1=2+Szn;i;j;k�1=2h= Gx� + cGxx + c2 nGyy � Syy +Gzz � Szzo+ch2�112 Gxxxx| {z }a1 +ch2 3�12 Gxxxx| {z }a2 +ch2�2�212 Gxxxx| {z }a3+ch2 148 n(Gyyyy � Syyyy) + (Gzzzz � Szzzz)o| {z }a4+ch2�4�248 n(Gyyyy � Syyyy) + (Gzzzz � Szzzz )o| {z }a5+ch2�116 n(Gyxxy � Syxxy) + (Gzxxz � Szxxz)o| {z }a6+ch2 2�16 n(Gyxxy � Syxxy) + (Gzxxz � Szxxz)o| {z }a7+ch2��212 n(Gxxzz � Sxxzz) + (Gxxyy � Sxxyy) + (Gyyzz � Syyzz)o| {z }part of a8+ch2��212 n(Gzyyz � Szyyz ) + (Gyxxy � Syxxy) + (Gzxxz � Szxxz)o| {z }part of a8+ch2 �8 n(Gyxyy + Syxyy) + (Gzxzz + Szxzz)o| {z }a9+ch2��212 n(Gyxyy + Syxyy) + (Gzxzz + Szxzz)o| {z }a10+O(h4) (5.4)for Gx, where all derivatives are evaluated at the cell center indexed (n; i; j; k). Asbefore, a fourth-order extension can be formulated by resolving the second-orderterm in the equivalent di�erential equation.At this point, it is important to note that none of the derivatives in a9 or a10of (5.4) can be discretized using � because data is not available where � requiresit. For example, Gyxyy � �xyyn;i;j;kGy cannot be used because all the data required by�xyyn;i;j;k are on x-faces, but Gy is kept only on y-faces.This problem can be circumvented either by the alternative �nite-di�erence oper-ator e� (which is a slightly di�erent but not novel �nite di�erence operator, de�ned in

124Appendix D) or by substituting in physically equivalent variables when the speci�edcharacteristic variables are absent. Using the e� when necessary, the second-ordererror terms in (5.4) can be discretized, giving the fourth-order scheme with theequivalent di�erential equationGxn+1;i+1=2;j;k�Gxn;i+1=2;j;k+Gxn;i�1=2;j;k�Gxn�1;i�1=2;j;k2�� + cGxn;i+1=2;j;k�Gxn;i�1=2;j;kh+ c2 Gyn;i;j+1=2;k�Gyn;i;j�1=2;k�Syn;i;j+1=2;k+Syn;i;j�1=2;kh+ c2 Gzn;i;j;k+1=2�Gzn;i;j;k�1=2�Szn;i;j;k+1=2+Szn;i;j;k�1=2h�ch2�112 �xxxn;i;j;kGx| {z }a1 �ch2 3�12 �xxxn;i;j;kGx| {z }a2 �ch2�2�212 �xxxn;i;j;kGx| {z }a3�ch2 148 n(�yyyn;i;j;kGy ��yyyn;i;j;kSy) + (�zzzn;i;j;kGz ��zzzn;i;j;kSz)o| {z }a4�ch2�4�248 n(�yyyn;i;j;kGy ��yyyn;i;j;kSy) + (�zzzn;i;j;kGz ��zzzn;i;j;kSz)o| {z }a5�ch2�116 n(�xxyn;i;j;kGy ��xxyn;i;j;kSy) + (�xxzn;i;j;kGz ��xxzn;i;j;kSz)o| {z }a6�ch2 2�16 n(�xxyn;i;j;kGy ��xxyn;i;j;kSy) + (�xxzn;i;j;kGz ��xxzn;i;j;kSz)o| {z }a7�ch2��212 n(�xzzn;i;j;kGx ��xzzn;i;j;kSx) + (�xyyn;i;j;kGx ��xyyn;i;j;kSx)o| {z }part of a8�ch2��212 n(�yzzn;i;j;kGy ��yzzn;i;j;kSy) + (�yyzn;i;j;kGz ��yyzn;i;j;kSz)o| {z }part of a8�ch2��212 n(�xxyn;i;j;kGy ��xxyn;i;j;kSy) + (�xxzn;i;j;kGz ��xxzn;i;j;kSz)o| {z }part of a8�ch2 �8 n( e�xyyn;i;j;kGy + e�xyyn;i;j;kSy) + ( e�xzzn;i;j;kGz + e�xzzn;i;j;kSz)o| {z }a9�ch2��212 n( e�xyyn;i;j;kGy + e�xyyn;i;j;kSy) + ( e�xzzn;i;j;kGz + e�xzzn;i;j;kSz)o| {z }a10= Gx� + cGxx + c2 nGyy � Syy +Gzz � Szzo+O(h4) (5.5)for Gx. We designate this scheme \ul3dAd-O4-A".

125For an alternative, we note that pressure p is kept on all faces as any of thecombinations p = Gx + Sx = Gy + Sy = Gz + Sz (5.6)Terms a9 and a10 in (5.4) can be rewritten using the equivalencen(Gyxyy + Syxyy) + (Gzxzz + Szxzz)o = n(Gxxyy + Sxxyy) + (Gxxzz + Sxxzz)o (5.7)switching from y-face and z-face data to x-face data so that the � operator canbe used. This alternative allows � to work on all terms, yielding the equivalentdi�erential equationGxn+1;i+1=2;j;k�Gxn;i+1=2;j;k+Gxn;i�1=2;j;k�Gxn�1;i�1=2;j;k2�� + cGxn;i+1=2;j;k�Gxn;i�1=2;j;kh+ c2 Gyn;i;j+1=2;k�Gyn;i;j�1=2;k�Syn;i;j+1=2;k+Syn;i;j�1=2;kh+ c2 Gzn;i;j;k+1=2�Gzn;i;j;k�1=2�Szn;i;j;k+1=2+Szn;i;j;k�1=2h�ch2�112 �xxxn;i;j;kGx| {z }a1 �ch2 3�12 �xxxn;i;j;kGx| {z }a2 �ch2�2�212 �xxxn;i;j;kGx| {z }a3�ch2 148 n(�yyyn;i;j;kGy ��yyyn;i;j;kSy) + (�zzzn;i;j;kGz ��zzzn;i;j;kSz)o| {z }a4�ch2�4�248 n(�yyyn;i;j;kGy ��yyyn;i;j;kSy) + (�zzzn;i;j;kGz ��zzzn;i;j;kSz)o| {z }a5�ch2�116 n(�xxyn;i;j;kGy ��xxyn;i;j;kSy) + (�xxzn;i;j;kGz ��xxzn;i;j;kSz)o| {z }a6�ch2 2�16 n(�xxyn;i;j;kGy ��xxyn;i;j;kSy) + (�xxzn;i;j;kGz ��xxzn;i;j;kSz)o| {z }a7�ch2��212 n(�xzzn;i;j;kGx ��xzzn;i;j;kSx) + (�xyyn;i;j;kGx ��xyyn;i;j;kSx)o| {z }part of a8�ch2��212 n(�yzzn;i;j;kGy ��yzzn;i;j;kSy) + (�yyzn;i;j;kGz ��yyzn;i;j;kSz)o| {z }part of a8�ch2��212 n(�xxyn;i;j;kGy ��xxyn;i;j;kSy) + (�xxzn;i;j;kGz ��xxzn;i;j;kSz)o| {z }part of a8�ch2 �8 n(�xyyn;i;j;kGx +�xyyn;i;j;kSx) + (�xzzn;i;j;kGx +�xzzn;i;j;kSx)o| {z }a9

126�ch2��212 n(�xyyn;i;j;kGx +�xyyn;i;j;kSx) + (�xzzn;i;j;kGx +�xzzn;i;j;kSx)o| {z }a10= Gx� + cGxx + c2 nGyy � Syy +Gzz � Szzo+O(h4) (5.8)for Gx which is only slightly di�erent from (5.5). We designate this scheme \ul3dAc-O4-B".The strategy of replacing the pressure of one face type by the pressure of anothersuggests that many schemes can be derived by similar substitutions or combinationsof similar substitutions, and a few remarks about this is appropriate. To minimizespurious anisotropy and asymmetry between coordinate directions, all coordinatesmust be treated symmetrically. In addition, E and H must also be treated symmet-rically. Within the equivalent di�erential equations, terms that are similar (havingcropped up from similar sources in the derivation of the equivalent di�erential equa-tion) must be discretized in the same manner. In the present case, this means thatall four derivatives in each of the terms a9 and a10 of equation (5.4) must undergosimilar substitutions. For example, carrying out the substitution for the �rst pairof derivatives in terms in a9 but not the second is not allowed, because they di�eronly by a coordinate rotation. Also not allowed is carrying out substitutions for Gbut not for S , because G and S di�er only by the sign of bn. At the expense ofconciseness, the truncation errors are grouped into numerous terms in the equivalentdi�erential equations to emphasize when they must undergo similar substitutionsand when their discretizations may di�er. The terms a9 and a10 are actually equiva-lent physical quantities, but they may be discretized di�erently, as are the group ofterms fa1; a2; a3g, fa4; a5g and fa6; a7g. In schemes ul3dAc-O4-A and ul3dAc-O4-B,a9 and a10 are discretized similarly, but they need not be, as we will show below in

127more variations on this scheme. Each discrete equation can be made more conciseby factoring its terms. This would be done in any implementations to avoid wastedrecomputations.The stencils required to discretize all six bicharacteristic equations are shown inFigure 5.4. The stencil for all equations are shown as one to emphasize that muchof the computations for the updates appear multiple times in the equations, andsigni�cant computational costs can be saved by not recomputing them each timethey are used. Note that both fourth-order stencils in Figure 5.4 are contained in a3x3 cell cube, but the second stencil does not involve the faces touching the middleof the edges of the 3x3 cube.As previously mentioned, the terms a9 and a10 in equation (5.4) need not bediscretized in identical manners. In ul3dAc-O4-A, they were discretized using the e�operator, and in ul3dAc-O4-B, they were exchanged via (5.7) for equivalent quantitiesthat can use the � operator. In a schemedesignated \ul3dAc-O4-C", the substitution(5.7) is used on term a10 but not term a9. And in a scheme designated \ul3dAc-O4-D", the substitution (5.7) is used on term a9 but not term a10. Schemes ul3dAc-O4-Cand ul3dAc-O4-D use the same stencil that ul3dAc-O4-A uses, but not in exactlythe same way. Since both a9 and a10 must both be computed, scheme ul3dAc-O4-Aand ul3dAc-O4-B involve the fewest oating point operations, since they are stillidentical after discretization.The stability limit is �max = 13 for all four ul3dAc-O4 schemes. The variations of�p over propagation angles are presented in Figures 5.5 for version A, 5.6 for versionB, 5.7 for version C and 5.8 for version D. The variations of �p are similar and theswitch of pressure seems to have no signi�cant e�ect on how the errors degrade forlow �. The worst error remains along the axes, and the error for propagation along

128ul3dAc-O2 ul3dAc-O4-A, B, Dul3dAc-O4-BStencil to update all characteristic variables interacting in center cell.Requires all characteristic variables on faces depicted.

xyzFigure 5.4: Stencils for ul3dAc schemes.

129the axes degrades faster than the error along the skewed angle as � is lowered.0 15 30 45

015

3045-0.01

00.010.02

� �� = �max� = 12�max

Figure 5.5: Phase error plots for scheme ul3dAc-O4-A.0 15 30 45

015

30450

0.010.02 � �

� = �max� = 12�maxFigure 5.6: Phase error plots for scheme ul3dAc-O4-B.The range of �p for discrete values of N and � are presented in Figures 5.3 forversion A, 5.4 for version B, 5.5 for version C and 5.6 for version D. Although the

1300 15 30 45

015

3045-0.01

00.010.02

� �� = �max� = 12�max

Figure 5.7: Phase error plots for scheme ul3dAc-O4-C.0 15 30 45

015

3045

0.01

0.02 � �� = �max� = 12�max

Figure 5.8: Phase error plots for scheme ul3dAc-O4-D.

131ranges of error are di�erent between them, the fastest (and usually worst) errors andthe propagation angle associated with them are identical; not surprisingly becausethe schemes di�er only in some multi-dimensional derivatives and the worst errors arefor 1D propagation along the axes. Schemes ul3dAc-O4-B and ul3dAc-O4-D, whichreplace a9 with an equivalent pressure term, have roughly half the anisotropic �p oful3dAc-O4-A and ul3dAc-O4-C, which did not replace a9. This can be con�rmedin Figure 5.9, which plots �p for ul3dAc-O4-C (best anisotropy) and ul3dAc-O4-D (worst anisotropy). Between the remaining ul3dAc-O4-B and ul3dAc-O4-D, thelatter has very slightly improved anisotropic �p than the former. This di�erence isdiminished at smaller Courant numbers. Interestingly, the a10 term is very similarto the a9 term, but it does not appear to contribute signi�cantly to the anisotropyof �p.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !-0.122 3.84!-0.407 2.98!-0.648 2.26!-1.01 1.12 5.575.264.924.12 7.657.256.815.75ul3dAc-O4-A (5.5) �max = 13Table 5.3: Error ranges for scheme ul3dAc-O4-A.In the above four variations of ul3dAc-O4, the e� operator is used only when thearrangement of discrete independent variables prevents the use of the � operator.This is natural. The greater compactness of the � stencils are more desirable becausethey simplify to some extent the logistics of coding, parallelizability and boundaryconditions. It is possible to substitute for the other terms in the equivalent di�er-

132��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !0.256 3.84!0.124 2.98!0.0174 2.26!-0.114 1.12 5.575.264.924.12 7.657.256.815.75ul3dAc-O4-B (5.8) �max = 13Table 5.4: Error ranges for scheme ul3dAc-O4-B.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !-0.152 3.84!-0.473 2.98!-0.765 2.26!-1.27 1.12 5.575.264.924.27 7.657.256.816.03ul3dAc-O4-C �max = 13Table 5.5: Error ranges for scheme ul3dAc-O4-C.ential equations, leading to larger stencils and more complexity. Any term that canbe discretized by � can be substituted and discretized using e�. Although unknownat this point, such substitution may lead to bene�ts such as increased stability limitand higher accuracy and, as we have seen, improved isotropy. An investigation intothis may be fruitful, however it is di�cult to determine a priori which of the contin-uously varying modi�cations will work. Because the present scheme works withoutany di�culty, there is little motivation at this point to explore these possibilities.Unfortunately, this is not true for the electromagnetic equations, as shown below.

133��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !0.286 3.84!0.19 2.98!0.133 2.26!0.142 1.12 5.575.264.924.12 7.657.256.815.75ul3dAc-O4-D �max = 13Table 5.6: Error ranges for scheme ul3dAc-O4-D.C

0 15 30 450

1530

45-0.015-0.01

-0.0050

0.0050.01

0.015D� �N = 4, � = �max

C

0 15 30 450

1530

450

0.010.02

D� �N = 4, � = 12�maxFigure 5.9: Comparison of �p between ul3dAc-O4-C and ul3dAc-O4-D5.2 Three-Dimensional ElectromagneticsThis section presents the fourth- and sixth-order extensions to the second-orderstandard leapfrog scheme and several versions of the fourth-order upwind leapfrogscheme for electromagnetics.5.2.1 Standard leapfrogFor the second-order standard leapfrog scheme for electromagnetics, the equiva-lent di�erential equation isEzn+1;i;j;k�Ezn�1;i;j;k2�� � cHxn;i;j+1=2;k�Hxn;i;j�1=2;k�Hyn;i+1=2;j;k+Hyn;i�1=2;j;kh

134= Ez� � c(Hxy �Hyx )+ch2 124 nHyxxx �Hxyyyo+ ch2 4�224 nHxyyy �Hyxxx +Hxxxy �Hyxyy +Hxyzz �Hyxzzo+O(h4) (5.9)for Ez, and those for the other components are similar. The procedure of resolvingthe second-order error term results in a fourth-order scheme with the equivalentdi�erential equationEzn+1;i;j;k�Ezn�1;i;j;k2�� � cHxn;i;j+1=2;k�Hxn;i;j�1=2;k�Hyn;i+1=2;j;k+Hyn;i�1=2;j;kh+ch2 124 n�xxxn;i;j;kHy ��yyyn;i;j;kHxo+ch2 4�224 n�yyyn;i;j;kHx ��xxxn;i;j;kHyo+ch2 4�224 n�xxyn;i;j;kHx ��xyyn;i;j;kHy +�yzzn;i;j;kHx ��xzzn;i;j;kHyo= Ez� � c(Hxy �Hyx )+ch4 27�120�2+48�45760 nHxyyyyy �Hyxxxxxo+ch4 80�2�48�45760 n�Hxyzzzz �Hxxxxxy +Hyxzzzz +Hyxyyyyo+ch4 96�45760 nHxxxyzz �Hyxyyzzo+ch4 96�4�40�25760 nHxxxyyy +Hxyyyzz �Hyxxxyy �Hyxxxzzo+O(h6) (5.10)for Ez. This scheme is designated \sl3dEm-O4". The stencils used for this schemeare shown in Figure 5.10c for a face-centered component and 5.10d for a line-centeredcomponent. This scheme is the same as that reported by Deveze [27]. Resolving thefourth-order error terms in equation (5.10) gives a sixth-order extension to Yee'sscheme. The stencils of Figures 5.10e and f are used for resolving these terms. Theextension gives the equivalent di�erential equationEzn+1;i;j;k�Ezn�1;i;j;k2�� � cHxn;i;j+1=2;k�Hxn;i;j�1=2;k�Hyn;i+1=2;j;k+Hyn;i�1=2;j;kh

135+ch2 124 n�xxxn;i;j;kHy ��yyyn;i;j;kHxo+ch2 4�224 n�yyyn;i;j;kHx ��xxxn;i;j;kHyo+ch2 4�224 n�xxyn;i;j;kHx ��xyyn;i;j;kHy +�yzzn;i;j;kHx ��xzzn;i;j;kHyo�ch4 27�120�2+48�45760 nHxyyyyy �Hyxxxxxo�ch4 80�2�48�45760 n��yzzzzn;i;j;kHx ��xxxxyn;i;j;kHx +�xzzzzn;i;j;kHy +�n;i;j;kHyo�ch4 96�45760 n�xxyzzn;i;j;kHx ��xyyzzn;i;j;kHyo�ch4 96�4�40�25760 n�xxyyyn;i;j;kHx +�yyyzzn;i;j;kHx ��xxxyyn;i;j;kHy ��xxxzzn;i;j;kHyo= Ez� � c(Hxy �Hyx )+O(h6) (5.11)for Ez. This scheme is designated \sl3dEm-O6". The stencils used for this schemeare shown in Figure 5.10e for the face-centered components and 5.10f for the line-centered components.Figures 5.11 and 5.12 shows how �p varies for discretizations of N = 4 at � = �maxand � = 12�max. Like the second- and fourth-order versions, the worst dispersion erroris along an axis and the best is along the most skewed angle. The stability limit isstill the same (�max = 12p3) and the errors remain on the slow side for all resolutionsand all � at all propagation angles.Tables 5.7 and 5.8 show the �p ranges for N = 4 and the resolution required toget one percent or 1360 error. Not too surprisingly, because these schemes share thesame stability limit as their counterparts for the acoustics problem, their results areidentical.

136sl3dEm-O2 face component stencil sl3dEm-O2 line component stencilsl3dEm-O4 face component stencil sl3dEm-O4 line component stencilsl3dEm-O6 face component stencil sl3dEm-O6 line component stencilStencil to update vertical �eld com-ponent on center face. Requireslongitudinal �eld component on alllines depicted. Stencil to update vertical �eld com-ponent along center line. Requiresnormal �eld component on all facesdepicted.

xyz

Figure 5.10: Stencils for sl3dEm schemes.

1370 15 30 45

015

3045

-0.02-0.015-0.01

-0.0050

� �� = �max� = 12�max

Figure 5.11: Phase error plots for scheme sl3dEm-O4.0 15 30 45

015

3045

-0.006-0.004-0.002

0

� �� = �max� = 12�max

Figure 5.12: Phase error plots for scheme sl3dEm-O6.

138��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !-2.39 -0.268!-2.29 -0.226!-2.16 -0.167!-1.75 0 5.044.984.894.61 7.016.926.796.38sl3dEm-O4 (5.10) �max = 12p3Table 5.7: Error ranges for scheme sl3dEm-O4.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !-0.753 -0.031!-0.722 -0.026!-0.677 -0.0191!-0.546 0 3.83.773.723.59 4.794.754.694.51sl3dEm-O6 (5.11) �max = 12p3Table 5.8: Error ranges for scheme sl3dEm-O6.5.2.2 Upwind leapfrogHigh-order extensions of the upwind leapfrog scheme for electromagnetics pro-ceeds by the same steps as for acoustics. However, there are some major di�cultiesthat arise here.The second-order scheme (4.9) yields the equivalent di�erential equationGxyn+1;i+1=2;j;k�Gxyn;i+1=2;j;k+Gxyn;i�1=2;j;k�Gxyn�1;i�1=2;j;k2�� + cGxyn;i+1=2;j;k�Gxyn;i�1=2;j;kh+ c2 �Gyxn;i;j+1=2;k+Syxn;i;j+1=2;k���Gyxn;i;j�1=2;k+Syxn;i;j�1=2;k�h+ c2 ��Gzyn;i;j;k+1=2+Szyn;i;j;k+1=2����Gzyn;i;j;k�1=2+Szyn;i;j;k�1=2�h

139= Gxy� + cGxyx � c2 nGyxy + Syxy �Gzyz + Szyz o+ch2�112 Gxyxxx| {z }a1 +ch2 3�12 Gxyxxx| {z }a2 +ch2�2�212 Gxyxxx| {z }a3+ch2�148 n(Gyxyyy + Syxyyy) + (�Gzyzzz + Szyzzz )o| {z }a4+ch2 4�248 n(Gyxyyy + Syxyyy) + (�Gzyzzz + Szyzzz )o| {z }a5+ch2 �212 n�(Gxyxyy + Sxyxyy) + (�Gxyxzz + Sxyxzz)o| {z }a6+ch2 348 n(Gyxxxy + Syxxxy) + (�Gzyxxz + Szyxxz)o| {z }a7+ch2�6�48 n(Gyxxxy + Syxxxy) + (�Gzyxxz + Szyxxz)o| {z }a8+ch2 4�248 n(Gyxxxy + Syxxxy) + (�Gzyxxz + Szyxxz)o| {z }a9+ch2 �8 n(�Gyxxyy + Syxxyy) + (Gzyxzz + Szyxzz)o| {z }a10+ch2 �212 n�(Gzyxzz + Szyxzz) + (Gzxyzz + Szxyzz)o| {z }a11+ch2 �212 n(Gyzyyz � Syzyyz) + (Gyxxyy � Syxxyy)o| {z }a12+ch2�3�24 n(Gyzxyz + Syzxyz) + (Gzxxyz � Szxxyz)o| {z }a13+ch2��26 Sxzxyz| {z }a14 +ch2 �212 n(Gyzxyz + Syzxyz) + (Gzxxyz � Szxxyz)o| {z }a15+O(h4) (5.12)for Gxy.Unlike the acoustics problem in which only p appears on more than one type offace (as G + S), here each of the six components of E and H appear on two facetypes so that equivalences similar to (5.6) can be written for any dependent variable.All innermost combinations of G and S in (5.12) reduce to one of the six componentsof the normalized �eld vectors that can be found on alternate faces. For example,

140a4 = (Gyxyyy + Syxyyy) + (�Gzyzzz + Szyzzz ) = Exyyy + Hxzzz . Then Ex can be taken fromthe z-faces giving Exyyy = Gzxyyy + Szxyyy and Hx can be taken from the y-faces givingHxzzz = Gyzzzz �Syzzzz . (Taking Ex from the y-faces and Hx from the z-faces gives backthe original expression.) Terms in which G or S appears alone can also be replacedwith equivalent expressions. For example, a14 = Sxzxyz, which is stored on an x-face,can be replaced with a14 = 12(Ezxyz + Hyxyz), with Ez taken from a y-face, and Hytaken from an z-face. As it is written, terms a1 through a9 in (5.12) can be discretizedusing the � operator, and replacing them with equivalent expressions changes this.Terms a10, a11 and a12 must use the e� operator in their current form, but they canbe replaced by equivalent expressions that can use the � operator. Terms a13, a14and a15 must use the e� operator. They can be replaced by equivalent expressionsbut not by any that can use the � operator. However, one can show that a15 = 2a14so that the last two terms cancel out.As before in the acoustic case, symmetry demands that coordinate directions betreated equally. The components of E and H must also be treated equally, sinceneither is mathematically more special than the other. Although this scheme canhave many variations, there are three basic stencil con�gurations to consider, asshown by the three larger stencils in Figure 5.13. The stencil ul3dEm-O4-A is usedif the explicit error terms in (5.12) are discretized as is. The stencil can be mademore compact by replacing terms a10, a11 and a12 by the equivalent quantities thatuse the � operator. This results in the stencil ul3dEm-O4-B. The expanded stencilul3dEm-O4-C or larger ones may result if the terms that can be discretized by �are replaced by terms that must be discretized by e�, providing more exibility ofdiscretization.An interesting question is how the substitutions a�ect numerical behavior of the

141ul3dEm-O2 ul3dEm-O4-A, B, D

ul3dEm-O4-B Other versions?Stencil to update all characteristic variables interacting in center cell.Requires all characteristic variables on faces depicted.xyz

Figure 5.13: Stencils for ul3dEm schemes.

142scheme. Unfortunately, though many variations for the discretization of the errorterms in (5.12) are possible, all that have been tried so far show pervasive stabilityproblems. (An exhaustive search into all possible continuous variations is impracticalwithout some systematic guide to narrow down the search.) The scheme is de�nitelyunstable for � > 12 (the 2D stability limit) so for the purpose of presentation, �maxis taken to be 12 , although it should be strictly unde�ned, except for stable schemes.For many propagation angles, the schemes have a small but nevertheless positive �abut are nevertheless unstable for any � over most of the 3D propagation angles.Despite the stability problem with these schemes, it may be instructive to showsome of their behaviors. We designate scheme \ul3dEm-O4-A" the scheme thatdiscretizes a1 through a9 using � and the rest bya10 � �xn;i;j;k n e�yyn;i;j;k(�Gyx + Syx) + e�zzn;i;j;k(Gzy + Szy)oa11 � �xn;i;j;k n� e�zzn;i;j;k(Gzy + Szy) + e�zzn;i;j;k(Gzx + Szx)oa12 � �zn;x;y;z e�yyn;i;j;k(Gyz � Syz) + �xn;i;j;k e�yyn;i;j;k(Gyx � Syx)a13 = a15 � �yn;i;j;k e�xzn;i;j;k(Gyz + Syz) + �zn;i;j;k e�xyn;i;j;k(Gzx � Szx)a14 � �xn;i;j;k e�yzn;i;j;kSxz (5.13)It uses no substitution of physically equivalent variables. Its �p variation over prop-agation angles are shown in Figure 5.14. Numerical values of �p are given in Table5.9. The maximum �a (including excess ampli�cation for spurious modes) for thisscheme (Figure 5.15) shows that although it is unstable over most propagation di-rections, it is stable when either � = 0 or � = 0, which is the 2D version of thisscheme. (Two-dimensional schemes are discussed below.) It is interesting to notethat scheme ul3dEm-O4-A is stable for the plane de�ned by � = �. Other versionsare not.

1430 15 30 45

015

3045

-0.010

0.01

� �� = �max� = 12�max

Figure 5.14: Phase error plots for scheme ul3dEm-O4-A.ul3dEm-O4-A

0 15 30 450

1530

4500.0010.0020.0030.004 � �N = 4, � = �max

ul3dEm-O4-A

0 15 30 450

1530

450

0.0005

0.001 � �N = 4, � = 12�maxFigure 5.15: Excess ampli�cation for scheme ul3dEm-O4-A.

144��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !-0.434 2.98!-0.815 1.94!-1.12 1.12!-1.54 0 5.264.744.124.45 7.256.575.756.25ul3dEm-O4-A (5.13) �max = 12 (but unstable)Table 5.9: Error ranges for scheme ul3dEm-O4-A.An alternative scheme, \ul3dEm-O4-B", di�erent from ul3dEm-O4-A in that it�rst replaces a10, a11 and a12 by variables that can be completely discretized by themore compact � operator, so that they becomea10 = (Gxyxzz + Sxyxzz) + (Gxyxyy � Sxyxyy)� �xzzn;i;j;k(Gxy + Sxy) + �xyyn;i;j;k(Gxy � Sxy)a11 = (Gxyxzz + Sxyxzz)� (Gyxyzz + Syxyzz)� �xzzn;i;j;k(Gxy + Sxy)��yzzn;i;j;k(Gyx + Syx)a12 = (�Gzyyyz + Szyyyz)� (Gxyxyy � Sxyxyy)� �yyzn;i;j;k(�Gzy + Szy) ��xyyn;i;j;k(Gxy � Sxy) (5.14)The variation of �p over propagation angles are shown in Figure 5.16. Numericalvalues of �p are given in Table 5.10. The maximum �a (including excess ampli�cationfor spurious modes) for this scheme is shown in Figure 5.17.The third alternative scheme, \ul3dEm-O4-C", di�erent from ul3dEm-O4-A inthat a15 is replaced by a14 so that the last two error terms in (5.12) cancel. Thevariation of �p for ul3dEm-O4-C over propagation angles are shown in Figure 5.18.Numerical values of �p are given in Table 5.11. The maximum �a (including excess

1450 15 30 45

015

3045

0

0.01 � �� = �max� = 12�max

Figure 5.16: Phase error plots for scheme ul3dEm-O4-B.ul3dEm-O4-B

0 15 30 450

1530

4500.050.1

0.15 � �N = 4, � = �maxul3dEm-O4-B

0 15 30 450

1530

450

0.0005

0.001 � �N = 4, � = 12�maxFigure 5.17: Excess ampli�cation for scheme ul3dEm-O4-B.

146��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !-0.0936 2.98!-0.354 1.94!-0.558 1.12!-0.803 0 5.264.744.123.79 7.256.575.755.22ul3dEm-O4-B (5.14) �max = 12 (but unstable)Table 5.10: Error ranges for scheme ul3dEm-O4-B.ampli�cation for spurious modes) for this scheme is shown in Figure 5.19.0 15 30 45

015

3045

-0.010

0.01

� �� = �max� = 12�max

Figure 5.18: Phase error plots for scheme ul3dEm-O4-C.The fourth alternative scheme, \ul3dEm-O4-D", combines the replacements oful3dEm-O4-B and of ul3dEm-O4-C. Its variation of �p over propagation directionsare shown in Figure 5.20. Numerical values of �p are given in Table 5.10. Themaximum �a (including excess ampli�cation for spurious modes) for this scheme isshown in Figure 5.21.

147ul3dEm-O4-C

0 15 30 450

1530

4500.020.040.060.08 � �N = 4, � = �max

ul3dEm-O4-C

0 15 30 450

1530

4500.00020.00040.00060.0008 � �N = 4, � = 12�maxFigure 5.19: Excess ampli�cation for scheme ul3dEm-O4-C.

0 15 30 450

1530

45-0.005

00.005

0.010.015

� �� = �max� = 12�max

Figure 5.20: Phase error plots for scheme ul3dEm-O4-D.

148��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !-0.437 2.98!-0.823 1.94!-1.14 1.12!-1.61 0 5.264.744.144.49 7.256.575.756.25ul3dEm-O4-C �max = 12 (but unstable)Table 5.11: Error ranges for scheme ul3dEm-O4-C.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !-0.0955 2.98!-0.36 1.94!-0.572 1.12!-0.844 0 5.264.744.123.84 7.256.575.755.26ul3dEm-O4-D �max = 12 (but unstable)Table 5.12: Error ranges for scheme ul3dEm-O4-D.The �p for these schemes are much better than those for the acoustics problem.At its maximum �, scheme ul3dEm-O4-B gets less than one percent error for N = 4.As in the acoustic case, the worst errors are the same for the four versions. However,schemes ul3dEm-O4-B and ul3dEm-O4-D, which replaced terms a10, a11 and a12so they can be discretized using the � operator, have signi�cantly smaller (half)anisotropic �p. Alas, all are unstable.

149ul3dEm-O4-D

0 15 30 450

1530

4500.050.1

0.15 � �N = 4, � = �maxul3dEm-O4-D

0 15 30 450

1530

450

0.0005

0.001 � �N = 4, � = 12�maxFigure 5.21: Excess ampli�cation for scheme ul3dEm-O4-D.5.3 Two-Dimensional Acoustics and ElectromagneticsThe 2D schemes can easily be reduced from the 3D schemes. Regardless ofwhether the acoustics equations or the electromagnetics equations are reduced, theresults are the same upon applying the transformation (2.15) or (2.17). The di�erencebetween the 2D and the 3D schemes is that the 2D schemes often have di�erent (ex-tended) stability ranges. Because as we have seen, numerical errors tend to degradeas � is reduced, the 2D schemes usually have lower error. Table 5.13 summarizes thedi�erent stability limits for all schemes examined up to now.We note that although no stable high-order upwind leapfrog scheme was foundfor 3D electromagnetics, the schemes examined were stable for wave propagationin any of the coordinate planes; hence their 2D counterparts work. All 3D acousticschemes, when reduced to 2D, have higher stability limits. Two-dimensional versionsof the 3D acoustics or electromagnetics schemes are identical.Plots of the variation in �p over propagation angles � and also over phase angles� for all the 3D schemes presented in this chapter are presented here for the 2Dversions. The plots for the standard leapfrog schemes are shown in Figures 5.22

150 �maxScheme 1D 2D 3D AC 3D EMSLF O(h2) 12 12 1p2 12 1p3 12 1p3SLF O(h4) 12 12 1p2 12 1p3 12 1p3SLF O(h6) 12 12 1p2 12 1p3 12 1p3ULF O(h2) 1 12 13 12ULF O(h4) 1 12 13 �Table 5.13: Comparison of 1D, 2D and 3D stability limits.and 5.23. Those for the upwind leapfrog schemes are shown in Figures 5.24, 5.25,5.26 and 5.27. Numerical values are given in Tables 5.14 and 5.15 for the standardleapfrog schemes and 5.16, 5.17, 5.18 and 5.19 for the upwind leapfrog schemes.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !-2.35 -0.592!-2.21 -0.502!-2 -0.374!-1.36 0 5.014.924.784.32 6.976.846.635.94sl2dAc-O4 �max = 12p2Table 5.14: Error ranges for scheme sl2dAc-O4.Schemes ul2dAc-O4-B and ul2dAc-O4-D, whose 3D versions did quite well, alsodo amazingly well at their stability limits in 2D. Schemes ul2dAc-O4-A and ul2dAc-O4-C become suddenly very accurate for (exactly) N = 2 at any propagation angle,

15145

90135

180

015

3045

-0.2

-0.1

0

� �� = �max� = 12�max

Figure 5.22: Phase error plots for scheme sl2dAc-O4.45

90135

180

015

3045

-0.2

-0.1

0

� �� = �max� = 12�max

Figure 5.23: Phase error plots for scheme sl2dAc-O6.

15245

90135

180

015

3045

00.10.2

� �� = �max� = 12�max

Figure 5.24: Phase error plots for scheme ul2dAc-O4-A.45

90135

180

015

3045

0

0.1

0.2

� �� = �max� = 12�max

Figure 5.25: Phase error plots for scheme ul2dAc-O4-B.

15345

90135

180

015

3045

-0.10

0.10.2

� �� = �max� = 12�max

Figure 5.26: Phase error plots for scheme ul2dAc-O4-C.45

90135

180

015

3045

0

0.1

0.2

� �� = �max� = 12�max

Figure 5.27: Phase error plots for scheme ul2dAc-O4-D.

154��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !-0.74 -0.1!-0.693 -0.0846!-0.626 -0.0626!-0.421 0 3.783.743.673.44 4.784.724.624.3sl2dAc-O6 �max = 12p2Table 5.15: Error ranges for scheme sl2dAc-O6.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !-0.0703 2.98!-0.577 1.94!-0.969 1.12!-1.46 0 5.264.744.124.44 7.256.575.756.25ul2dAc-O4-A �max = 12Table 5.16: Error ranges for scheme ul2dAc-A-O4.at their stability limits, but this is hardly useful. As in 3D, these 2D versions, withthe exception of ul2dAc-O4-D at ��max = 1, have identical fast �a. At ��max = 1, theworst error is the slow error, and versions B and D, with substitutions that improvedthe worst errors, are clearly the best.These high-order 2D upwind leapfrog results indicate that these schemes are verypromising candidates for problems run with � = 12. In fact, at this Courant number,the schemes also simplify greatly and become very e�cient as many truncation errorterms drop out. For this Courant number, a fourth-order, 2D scheme can be derivedby resolving only a small number of remaining error terms resulting in a very e�cient

155��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !0.496 2.98!0.194 1.94!-0.0241 1.12!-0.22 0 5.264.744.122.68 7.256.575.753.76ul2dAc-O4-B �max = 12Table 5.17: Error ranges for scheme ul2dAc-B-O4.��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !-0.141 2.98!-0.732 1.94!-1.24 1.12!-2.08 0 5.264.744.254.89 7.256.576.036.87ul2dAc-O4-C �max = 12Table 5.18: Error ranges for scheme ul2dAc-C-O4.scheme for problems with uniform media.Reducing equation (5.4) to a 2D form, leavesGxn+1;i+1=2;j�Gxn;i+1=2;j+Gxn;i�1=2;j�Gxn�1;i�1=2;j2�� + cGxn;i+1=2;j�Gxn;i�1=2;jh+ c2 Gyn;i;j+1=2�Gyn;i;j�1=2�Syn;i;j+1=2+Syn;i;j�1=2h= Gx� + cGxx + c2 nGyy � Syyo+ch2�112 Gxxxx| {z }a1 +ch2 3�12 Gxxxx| {z }a2 +ch2�2�212 Gxxxx| {z }a3+ch2 148 n(Gyyyy � Syyyy)o| {z }a4 +ch2 (�4�248 n(Gyyyy � Syyyy)o| {z }a5 +ch2�116 n(Gyxxy � Syxxy)o| {z }a6

156��max 100�p(N = 4) N(�p = 1%) N(�p = 1360)1312231 !0.566 2.98!0.348 1.94!0.244 1.12!0 0.394 5.264.744.122.93 7.256.575.754.44ul2dAc-O4-D �max = 12Table 5.19: Error ranges for scheme ul2dAc-D-O4.+ch2 2�16 n(Gyxxy � Syxxy)o| {z }a7 +ch2��212 n(Gxxyy � Sxxyy) + (Gyxxy � Syxxy)o| {z }a8+ch2 �8 n(Gyxyy + Syxyy)o| {z }a9 +ch2��212 n(Gyxyy + Syxyy)o| {z }a10 +O(h4) (5.15)where the terms have been labeled the same names as those in 3D counterpartequation (5.4). At � = 12, this equation simpli�es to the equivalent di�erentialequation cGxn+1;i+1=2;j�Gxn�1;i�1=2;jh + c2 (Gyn;i;j+1=2�Syn;i;j+1=2 )�(Gyn;i;j�1=2�Syn;i;j�1=2 )h= Gx� + cGxx + c2 nGyy � Syyo+ch2�148 n(Gxxyy � Sxxyy) + (Gyxxy � Syxxy)o| {z }a8 +ch2 116 nGyxyy + Syxyyo| {z }a9+ch2�148 nGyxyy + Syxyyo| {z }a10 +O(h4) (5.16)For ul2dAc-O4-B, which makes the substitution (Gy + Gy) = (Gx + Gx) for a9 anda10, the equivalent di�erential equation becomescGxn+1;i+1=2;j�Gxn�1;i�1=2;jh + c2 (Gyn;i;j+1=2�Syn;i;j+1=2 )�(Gyn;i;j�1=2�Syn;i;j�1=2 )h= Gx� + cGxx + c2 nGyy � Syyo� ch248 n(Gxxyy � Sxxyy) + (Gyxxy � Syxxy)o+ ch224 nGxxyy + Sxxyyo+O(h4)

157grid linesdata used and updateddata used xyFigure 5.28: Stencil for scheme ul2dAc-O4-B at � = 12.= Gx� + cGxx + c2 nGyy � Syyo+ ch248 n4Sxxyy + (Gxxyy � Sxxyy)� (Gyxxy � Syxxy)o+O(h4) (5.17)The last equality (using no further substitution) gives the second-order truncationerror in terms of the reverse moving characteristic variable and the vorticity! = vx � uy (5.18)If the condition of irrotationality is imposed, the vorticity term can be disregardedwithout changing the physical meaning of the equation. However, failure to discretizethe vorticity leads to an unstable scheme.The second-order error terms in (5.17) can be uniquely resolved by a slight ex-tension of the original spatial stencil, as shown in Figure 5.28. Only the stencil atthe middle time level is extended. The data in the past and future steps remainidentical to that in Figure 4.2.

CHAPTER VIEFFICIENCYIt is important to note that the relationship between error and resolution is notby itself adequate to assess the cost of a simulation. The cost associated with agiven resolution is also an important factor to consider. We de�ne e�ciency asthe cost for a solution given an error requirement. E�ciency is de�ned for bothmemory required and CPU (central processing unit) time required to perform thesimulation. Quantitatively, memory requirement is given by the number of oating-point numbers that must be stored and time requirement is given by the number of oating-point operations ( ops) that must be performed. In this regard, the simplerschemes have the advantage since they require less storage and fewer oating pointoperations.E�ciency usually presents a road block in higher dimensions, since the compu-tational requirement grows exponentially with the dimension number. The physicaldomain of a problem can be characterized by its linear dimension, which is propor-tional to the wavelength (`). As in Shlager [46], we give the cost for a domain thatis one wavelength on each side. Domains larger or smaller than these will have totalcosts that di�er from our numbers by a �xed constant, thus the relative costs of eachscheme with respect to one another will still be the same. For integration time, we158

159choose one period. For problems with di�erent integration times, the cost will scalethe same manner, so relative costs remain unchanged.The error requirement determines the resolution N for each individual scheme.The 2D domain for this study requires N2 cells to discretize, and the 3D domainrequires N3 cells to discretize. The memory requirement will then be the number ofcells times the memory requirement of one cell. To integrate one period requires N�time steps, using the time step size corresponding to �. Thus the time cost of the2D problem is N2N� times the time cost of updating one cell one step. For the 3Dproblem it is N3N� times the time cost of updating one cell one step.It is clear that cost grows very quickly with resolution. It is interesting to notethat error also drops quickly with N , independent of the number of dimensions.Second order schemes reduces error in proportion to N�2, fourth order schemes inproportion to N�4, sixth order schemes in proportion to N�6, etc. For high-orderschemes, errors drop faster than the computational requirement as N increases, soit is advisable to increase N as well as the order of accuracy, when trying to reduceerror. The use of parallel computers may be required to handle the memory demands,which adds some impetus to using easily parallelizable algorithms.The required resolution for a given level of error has been given in previouschapters. In this chapter, the corresponding computational requirements are given.Tables 6.1, 6.2 and 6.3 give the per cell storage and op requirements for the 2Dacoustic schemes, 3D acoustics schemes and 3D electromagnetics schemes respec-tively. The op counts include all ops. All algebraic operations (add, subtract,multiply and divide) are counted equally. No other type of operations were involved.The higher memory requirement of the upwind leapfrog schemes are due to theirgreater complexity. The standard leapfrog schemes are very memory e�cient because

160they store only partial solutions at each level (for example, pressure or velocity, butnot both). The upwind leapfrog schemes store two full levels and also require morenumbers to be stored per level due to their speci�c storage strategies. High-orderschemes require no more storage per cell than their lower-order counterparts, butthey do require more ops to update one cell. Of the fourth-order upwind leapfrogschemes, version B, requires the fewest number of ops. This, along with the superiorerrors of version B make this version the best among the fourth-order upwind leapfrogschemes.We stress again that these numbers re ect only the bare essentials of the scheme.The memory requirement per cell re ect only the solution memory. It does not re ectother data (such as material property) that may be stored on the grid. The opsrequirement re ects only the computations required to update the solution. A ho-mogeneous medium was assumed. Inhomogeneous media will require more memoryas well as more computations. A reasonable e�ort was made to reduce the operationcounts per cell. Parameters that do not vary from cell to cell were lumped togetheras much as possible and stored to avoid unnecessary recomputations. We mentionedin Section 5.3 that upwind leapfrog schemes at � = 12 do not require certain com-putations that are required for a general Courant number, thus reducing their oprequirements. However, this advantage was not taken in the op counts presented.More complexity such as inhomogeneity would improve the total cost the schemesthat use fewer cells relative to those for schemes that uses more cells. For this rea-son, we also present the total number of cells required for each scheme (below). Thismeasure is called the \cell e�ciency".The resolutions required to attain given levels of error were given in the previouschapters. Using this data and the per cell requirements, the total cost (in terms of

161 oating point oating pointScheme storage per cell operations per cellsl2dAc-O2 3 5.5sl2dAc-O4 3 34.5sl2dAc-O6 3 112.5ul2dAc-O2 8 20ul2dAc-O4-A 8 136ul2dAc-O4-B 8 104ul2dAc-O4-C 8 144ul2dAc-O4-D 8 144Table 6.1: Per cell computational requirement for 2D acoustics and electromagneticsschemes. oating point oating pointScheme storage per cell operations per cellsl3dAc-O2 4 7sl3dAc-O4 4 71sl3dAc-O6 4 333ul3dAc-O2 12 36ul3dAc-O4-A 12 267ul3dAc-O4-B 12 219ul3dAc-O4-C 12 279ul3dAc-O4-D 12 279Table 6.2: Per cell computational requirement for 3D acoustics schemes.

162 oating point oating pointScheme storage per cell operations per cellsl3dEm-O2 6 15sl3dEm-O4 6 177sl3dEm-O6 6 645ul3dEm-O2 24 84Table 6.3: Per cell computational requirement for 3D electromagnetics schemes.storage and ops) of the simulation can be computed. In Tables 6.4, 6.5 and 6.6, wepresent the resulting numbers for the 2D and 3D schemes. The scheme names aregiven in the �rst column. The second column restates the resolution requirements toget �p = 1%. The third column shows N�max which is the number of time steps requiredat the maximum stable time Courant number (where each scheme is at its moste�cient and accurate). The fourth column labeled \cells" shows the total number ofcells required. The �fth column, labeled \mem" shows the memory requirement interms of the number of oating-point numbers stored. And the last column, labeled\ ops" shows the number of ops required to complete the integration.The numbers given in these tables may seem authoritative, so it is important toremember that they may yet be improved upon. Recall that no e�ort was made instudying the e�ects that alternative discretizations of the error terms may have onthe accuracy of the high-order schemes. Future e�orts may wish to investigate this.For now, we will examine what we have.In 2D (Table 6.4), the most cell e�cient is the ul2dAc-O4-B scheme and the leastis the sl2dAc-O2 scheme. For memory e�ciency, the best is the sixth-order standard

163 �p = 1%Scheme N N=�max cells mem opssl2dAc-O2 9.163 25.92 83.95 252 11967sl2dAc-O4 4.315 12.21 18.62 56 7841sl2dAc-O6 3.437 9.721 11.81 35 12916ul2dAc-O2 6.479 12.96 41.98 336 10879ul2dAc-O4-A 4.44 8.88 19.72 158 23811ul2dAc-O4-B 2.684 5.369 7.206 58 4023ul2dAc-O4-C 4.894 9.787 23.95 192 33752ul2dAc-O4-D 2.928 5.855 8.571 69 7226Table 6.4: E�ciency comparison for 2D acoustics schemes at �p = 1%�p = 1%Scheme N N=�max cells mem opssl3dAc-O2 10.52 36.44 1164 4655 296798sl3dAc-O4 4.613 15.98 98.15 393 111349sl3dAc-O6 3.588 12.43 46.2 185 191238ul3dAc-O2 8.55 25.65 625 7500 577084ul3dAc-O4-A 4.12 12.36 69.91 839 230700ul3dAc-O4-B 4.12 12.36 69.91 839 189226ul3dAc-O4-C 4.273 12.82 78.01 936 278995ul3dAc-O4-D 4.12 12.36 69.91 839 241068Table 6.5: E�ciency comparison, 3D acoustics schemes at �p = 1%

164 �p = 1%Scheme N N=�max cells mem opssl3dEm-O2 10.52 36.44 1164 6982 635995sl3dEm-O4 4.613 15.98 98.15 589 277589sl3dEm-O6 3.588 12.43 46.2 277 370409ul3dEm-O2 7.654 15.31 448.3 10760 576492Table 6.6: E�ciency comparison, 3D electromagnetics schemes at �p = 1%leapfrog scheme and the worst is the second order standard leapfrog scheme. Thememory requirements are similar for the fourth-order schemes sl2dAc-O4 and ul2dAc-O4-B. The best time e�ciency belongs to the ul2dAc-O4-B scheme and the worst tothe sl2dAc-O6 scheme (not counting the lesser ul2dAc-O4 schemes since they wouldnever be used anyway).Data for the 3D acoustic schemes (Table 6.5) show that scheme sl3dAc-O6 ismost cell-e�cient and memory-e�cient, but due to complexity, loses to the fourth-order sl3dAc-O4 for time e�ciency. The upwind leapfrog schemes for 3D acousticsdo not perform well in this instance, because of their lower �max penalizing both timestep size and amount of error accumulated per period. However, the second-orderscheme ul3dAc-O2 is more cell-e�cient than the second-order sl3dAc-O2 scheme. Allfourth-order upwind leapfrog schemes are signi�cantly better than the second-orderstandard leapfrog scheme in all three measures of e�ciency.Data for the 3D electromagnetic schemes (Table 6.6) shows that the second-orderupwind leapfrog scheme was able to improve on the second-order standard leapfrogscheme in terms of cell-e�ciency, but not memory and time e�ciency. The most

165memory-e�cient is the sixth-order sl3dEm-O6 scheme and the most time-e�cient isthe fourth-order sl3dEm-O4 scheme. However, in case the stencil of the sixth-orderscheme is too undesirable, the fourth-order scheme is still an enormous improvementover the second-order standard leapfrog scheme.Tables 6.7, 6.8 and 6.9 show the e�ciency data, but with the error requirementtightened to �p = 1360. When �p drops like this, it always favors the high-orderschemes more than the low-order schemes. One major consequence of lowering theerror requirement is that the sixth-order sl3dEm-O6 is �nally better than the fourth-order sl3dEm-O6 in ops requirement. This is the only case of such pay o� though,in the two �p's we examined. This suggests that unless error requirements are fairlytight, one has little reason to resort to the sixth-order schemes, other than to gainthe memory e�ciency. �p = 1360Scheme N N=�max cells mem opssl2dAc-O2 17.26 48.81 297.8 893 79950sl2dAc-O4 5.939 16.8 35.27 106 20438sl2dAc-O6 4.3 12.16 18.49 55 25303ul2dAc-O2 12.2 24.41 148.9 1191 72682ul2dAc-O4-A 6.254 12.51 39.11 313 66531ul2dAc-O4-B 3.764 7.527 14.17 113 11089ul2dAc-O4-C 6.868 13.74 47.17 377 93311ul2dAc-O4-D 4.439 8.878 19.71 158 25192Table 6.7: E�ciency comparison for 2D acoustics schemes at �p = 1360

166 �p = 1360Scheme N N=�max cells mem opssl3dAc-O2 19.89 68.91 7873 31493 3798122sl3dAc-O4 6.38 22.1 259.7 1039 407559sl3dAc-O6 4.507 15.61 91.57 366 476108ul3dAc-O2 16.21 48.64 4262 51144 7462833ul3dAc-O4-A 5.749 17.25 190 2280 875111ul3dAc-O4-B 5.749 17.25 190 2280 717788ul3dAc-O4-C 6.029 18.09 219.2 2630 1105938ul3dAc-O4-D 5.749 17.25 190 2280 914442Table 6.8: E�ciency comparison, 3D acoustics schemes at �p = 1360�p = 1360Scheme N N=�max cells mem opssl3dEm-O2 19.89 68.91 7873 47240 8138833sl3dEm-O4 6.38 22.1 259.7 1558 1016028sl3dEm-O6 4.507 15.61 91.57 549 922191ul3dEm-O2 14.18 28.37 2853 68477 6798559Table 6.9: E�ciency comparison, 3D electromagnetics schemes at �p = 1360

167To show where the pay o� points lie for these problems, Figures 6.1, 6.2 and 6.3plot the total memory and ops requirements as a function of error requirement, forerror requirements from one-tenth of one percent to ten percents. Some lines, such asthat of scheme ul2dAc-O2 (Figure 6.1), change slope near the high error end becausethe second-lowest-order term in the truncation error has begun to dominate overthe lowest-order term. The lines for the 2D fourth-order upwind leapfrog, versionB scheme (Figure 6.1) level out at the high error end, indicating that the lowestallowable resolution (N = 2) has been reached. Resolution is not allowed to gobelow this number due to aliasing. The most e�cient schemes are represented bythe lowest lines and the least e�cient are represented by the highest lines. Thepay-o� points occur when the lines cross.For memory e�ciency, the sixth-order standard leapfrog schemes are always thebest. This is expected because the standard leapfrog schemes are very memorye�cient and memory e�ciency always improve with the order of accuracy. For ope�ciency, these complex schemes loose ground to the simpler schemes that use fewer ops per cell update.In 2D, scheme ul2dAc-O4-B is the most op e�cient for the error range in[:001; :036] and schemeul2dAc-O2 scheme is most e�cient for the error range [:036; :1].For 3D acoustics, scheme sl3dAc-O6 is most op e�cient in the range [:001; :0016],scheme sl3dAc-O4 in the range [:0016; :017] and sl3dAc-O2 in the range [:017; :1]. For3D electromagnetics, scheme sl3dEm-O6 is most op e�cient in the range [:001; :0036],scheme sl3dEm-O4 in the range [:0036; :012] and sl3dEm-O2 in the range [:012; :1].While these comparisons are very useful, they are restricted to certain classesof problems as noted in [46], and can change slightly or drastically as more generalproblems are considered. For example, an inhomogeneous problem requires consti-

16810

100

1000

10000

0.001 0.01 0.1

Mem

ory

requ

irem

ent

error

sl2dAc-O2sl2dAc-O4sl2dAc-O6ul2dAc-O2

ul2dAc-O4-B

100

1000

10000

100000

1e+06

0.001 0.01 0.1

Flo

ps r

equi

rem

ent

error

sl2dAc-O2sl2dAc-O4sl2dAc-O6ul2dAc-O2

ul2dAc-O4-B

Figure 6.1: Computational resource requirements for 2D acoustics schemes.

16910

100

1000

10000

100000

1e+06

0.001 0.01 0.1

Mem

ory

requ

irem

ent

error

sl3dAc-O2sl3dAc-O4sl3dAc-O6ul3dAc-O2

ul3dAc-O4-B

1000

10000

100000

1e+06

1e+07

1e+08

0.001 0.01 0.1

Flo

ps r

equi

rem

ent

error

sl3dAc-O2sl3dAc-O4sl3dAc-O6ul3dAc-O2

ul3dAc-O4-B

Figure 6.2: Computational resource requirements for 3D acoustics schemes.

17010

100

1000

10000

100000

1e+06

0.001 0.01 0.1

Mem

ory

requ

irem

ent

error

sl3dEm-O2sl3dEm-O4sl3dEm-O6ul3dEm-O2

1000

10000

100000

1e+06

1e+07

1e+08

0.001 0.01 0.1

Flo

ps r

equi

rem

ent

error

sl3dEm-O2sl3dEm-O4sl3dEm-O6ul3dEm-O2

Figure 6.3: Computational resource requirements for 3D electromagnetics schemes.

171tutive parameters to be stored throughout the grid, giving an advantage to schemesthat use coarser grids. (The standard leapfrog storage scheme is particularly poor inthis regard because its di�erence stencils are centered at a di�erent place for everyvariable updated, and the constituent parameters must be stored for each stencil.The upwind leapfrog scheme need only store constituent parameters at the cell cen-ters.) Extensibility for more complex problems is another issue which has strongimplications when it must be considered. Interactions between the �elds, other thanwhat is set up by the basic schemes may not be be possible with certain schemes.The impedance boundary condition would not be straight forward using the standardleapfrog scheme (see Appendix B). Chiral media are not possible with the standardleapfrog scheme, but may be possible with the upwind leapfrog schemes. Anisotropicmedia in which the principle axes not coinciding with the coordinate axes are notpossible with either scheme in their current forms.Past the numerics of a scheme, the implementation on a computer can makea di�erence in the desirability of a scheme. Ease of use, especially at boundaryconditions and in domain decomposition, favor small, compact stencils, such as thatof the second-order upwind leapfrog schemes. High-order schemes' stencils reachingout in all directions can increase communication overhead in by requiring data froma greater number of partitions in a parallel computation.Because the requirements may vary considerably from one problem to the next,there is no overall \best buy", but the analysis in this chapter will provide someinitial guidance. The choice of numerical schemes should be based in part on thatthe problem requires and what computer resources are necessary.

CHAPTER VIINUMERICAL EXPERIMENTSThe mathematical predictions shown in the previous chapters are very helpful inpredicting the performance of numerical schemes. However, they do not say every-thing. In this chapter, some simple numerical results are presented that shed morelight on the performances of these schemes.Fairly simple problems with exact solutions are chosen to test the schemes. Sincethese are linear schemes, more complex solutions can be thought of as superpositionsof the simple solutions presented and are expected to behave similarly. Choosingsimple problems allows us to draw more conclusions about the basic strengths andweaknesses of the schemes, free from complex superpositions of many waves withmany di�erent error characteristics. Solutions of a numerical scheme are comparedwith that of others and with the exact solutions. The presentation of numericalsolutions and formal analyses together in this dissertation provides two angles fromwhich to assess the numerical schemes.In 1D, the linear advection of a pulse and sinusoidal waves are chosen. In higherdimensions, the problem of a point source (pressure in the acoustic problem, dipolein the the electromagnetic problem) is chosen.172

173-0.4-0.2

00.20.40.60.8

11.2

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.1: 1D pulse advection result using scheme sl1dAd-O2.7.1 One-Dimensional AdvectionNumerical experiments for 1D advection include running a continuous wave anda pulse (a single cycle of a wave with the same frequency). In each experiment, theinitial solution is set to the exact solution, then the wave is propagated a number ofcycles. The domain is x 2 [0; 1], divided into 32 cells. The wave from which the pulseis taken has a wavelength of 14, corresponding to a resolution of N = 8. (The pulseitself is broad-band, of course.) The waves are propagated to � = 32, equivalent to128 periods. The Courant number of � = 13 was used on all schemes, even those with�max = 1 because we have in mind the possible development of multi-dimensionalschemes by adding cross terms to these schemes. In addition it appears unlikelybased on the results of Chapters IV and V that the multi-dimensional schemes willhave this high a stability limit. This choice of Courant numbers favors the schemeswith lower stability limits, since accuracy degrades when � is lowered.The results for the 1D pulse advection are shown in Figures 7.1 through 7.14for the pulse and Figures 7.15 through 7.28 for the continuous wave. They arepresented roughly in order of their degree of accuracy. In general, the numericalresults corroborate the formal predictions. But there are some interesting surprises.The performances of the second-order accurate schemes (Figures 7.1 and 7.2) seem

174-0.4-0.2

00.20.40.60.8

11.2

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.2: 1D pulse advection result using scheme ul1dAd-O2.

-0.4-0.2

00.20.40.60.8

11.2

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.3: 1D pulse advection result using scheme sl1dAd-O4.

-0.4-0.2

00.20.40.60.8

11.2

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.4: 1D pulse advection result using scheme ul1dAd-O4.

-0.4-0.2

00.20.40.60.8

11.2

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.5: 1D pulse advection result using scheme sl1dAd-O6.

175-0.4-0.2

00.20.40.60.8

11.2

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.6: 1D pulse advection result using scheme ul1dAd-O6.

-0.4-0.2

00.20.40.60.8

11.2

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.7: 1D pulse advection result using scheme ciA.

-0.4-0.2

00.20.40.60.8

11.2

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.8: 1D pulse advection result using scheme ciB.

-0.4-0.2

00.20.40.60.8

11.2

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.9: 1D pulse advection result using scheme uiA.

176-0.4-0.2

00.20.40.60.8

11.2

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.10: 1D pulse advection result using scheme uiB.

-0.4-0.2

00.20.40.60.8

11.2

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.11: 1D pulse advection result using scheme uiC.

-0.4-0.2

00.20.40.60.8

11.2

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.12: 1D pulse advection result using scheme uiD.

-0.4-0.2

00.20.40.60.8

11.2

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.13: 1D pulse advection result using scheme wl5.

177-0.4-0.2

00.20.40.60.8

11.2

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.14: 1D pulse advection result using scheme wlC.particularly poor, and even the fourth-order schemes (Figures 7.3 and 7.4) are notmuch better. This should not be taken to be a death sentence for these schemes. Theconditions of the problem were a very rigorous combination of coarse resolution andlong integration time, chosen to di�erentiate the performances of the more accurateschemes. The schemes were not allowed to run at the maximumCourant number, forthat leads to zero error (in 1D) and no signi�cant conclusion. Second-order accurateschemes have the formidable advantage of presenting the least problem at boundariesand may prove very useful in hybrid schemes.For the same order of accuracy and the same resolution and � = 13, the upwindleapfrog schemes (Figures 7.2, 7.4 and 7.6) are more accurate than the standardleapfrog schemes (Figures 7.1, 7.3 and 7.5). However, recall that for lower �, thestandard leapfrog schemes takes the lead.For this broad-band problem with much high-frequency content, the di�usiveschemes seem to be better at maintaining the shape of the pulse than the non-di�usiveschemes do. For example, of the optimizable schemes (3.24), the least dispersive,most di�usive version (w = w5th) is best at maintaining the shape of the pulse, andthe most dispersive, least di�usive version (w = 0) is worst. The compromise (wCth)falls in between (Figures 7.4 7.13 and 7.14). The ability of the di�usive schemes tomaintain the shape of the pulse is attributed to the generally greater damping on

178-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.15: 1D wave advection result using scheme sl1dAd-O2.

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.16: 1D wave advection result using scheme sl1dAd-O4.the high-frequency errors. Left undamped it is the dispersion of these high-frequencycomponents that alter the shape of the waves. A further example can be seen bycomparing the highly di�usive scheme uiA to the non-di�usive schemes sl1dAd-O6and ul1dAd-O6. Scheme uiA has the worst phase speed error of the three, and muchworse di�usive errors, yet it has much less prominent wiggles outside of the pulse.This conclusion supports the di�usive approaches of Omick [40] and Shankar [31] overthe non-di�usive approaches of Yee and many others. Omick's model problem is thepropagation of a step function, having even stronger high-frequency components thanthe pulse used here. But before we can conclude that dispersion is more importantthan di�usion, let us examine the schemes' performance on the continuous wave.The results of the continuous wave problem immediately bring out a crucial weak-ness of di�usive schemes: the loss of wave strength despite very good phase errors.This is exempli�ed by schemes ciA (Figure 7.18), uiA (Figure 7.23) and uiC (Fig-

179-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.17: 1D wave advection result using scheme sl1dAd-O6.

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.18: 1D wave advection result using scheme ciA.

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.19: 1D wave advection result using scheme ciB.

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.20: 1D wave advection result using scheme ul1dAd-O2.

180-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.21: 1D wave advection result using scheme ul1dAd-O4.

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.22: 1D wave advection result using scheme ul1dAd-O6.

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.23: 1D wave advection result using scheme uiA.

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.24: 1D wave advection result using scheme uiB.

181-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.25: 1D wave advection result using scheme uiC.

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.26: 1D wave advection result using scheme uiD.

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.27: 1D wave advection result using scheme wl5.

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

exactnumericalxG Figure 7.28: 1D wave advection result using scheme wlC.

182-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

exactciBul6sl6xG Figure 7.29: 1D wave advection result using various schemes at N = 4.ure 7.25). It is visible to a lesser extend in scheme wl5 (Figure 7.27). Scheme wlC(Figure 7.28) is con�rmed to be a successful compromise between the best ampli-tude error version (Figure 7.21) and the best phase error version (Figure 7.27). Thisresults suggests that perhaps the best scheme for a problem would have to be afunction of what features are important in the problem. It is surprising to see thatsome di�usive schemes display virtually no amplitude error, despite predictions ofsigni�cant �a; in particular, scheme ciB. This scheme is rescued by the fact that itis has very high-order truncation errors. Recall that �a is seventh order accuratefor this scheme! The improvements in the order of accuracy of a scheme may notbring signi�cant improvement to the solution at a �xed resolution. The strength ofhigh-order schemes lies also in the dividend returned for investing more resolution.In Figure 7.29, the limits of scheme ciB is exposed by the same problem with half asmany points, bringing the resolution to N = 4. The solutions at � = 8 (32 periodsof integration) shows signi�cant loss of amplitude in the scheme ciB, while schemessl1dAc-O6 and ul1dAd-O6 are both better.7.2 Two-Dimensional In�nite Radiating CurrentWe consider the problem of the point pressure source of�(� ) = cos (!� + ') (7.1)

183applied to the 2D acoustic equations (2.14). This is equivalent to the problem of anin�nite oscillating current in a wire. This source is located at the spatial coordinates(0; 0). The exact solution for the pressure �eld ispex = ��4 [J0(�r) cos(!� + ') + Y0(�r) sin(!� + ')] (7.2)where r is the distance from the source, and J0 and Y0 are Bessel functions. Thenumerical scheme is the same as the source-free scheme, except where the sourceresides. For the standard leapfrog scheme, the source is placed at a cell center,coinciding with a pressure storage location, changing equation 4.1a topn+1;i;j � pn+1;i;j2�� + c ��xn;i;ju +�yn;i;jv�+�(n�� ) = 0 (7.3)For the upwind leapfrog scheme, the source is also placed at the cell center, changing,for example, equation (4.3)a toG+xn+1;i+1=2;j �G+xn;i+1=2;j +G+xn;i�1=2;j �G+xn�1;i�1=2;j2��+c�xn;i;jG+x + c2�yn;i;j �G+y �G�y�+ �(n�� )2 = 0 (7.4)The higher order di�erence equations follow similarly. The schemes are run at theirmaximum stable Courant numbers to maximize e�ciency and minimize errors.The numerical solution is initialized to zero before starting the integration. Theshift angle is set to ' = ��2 to avoid an initial discontinuity in the source. Disconti-nuities are undesirable because they can trigger strong, slow moving spurious modesthat linger and obscure the physical modes. The grids used were su�ciently large sothat the performance of absorbing boundary conditions does not raise uncertaintiesregarding the source of the errors. Finally, symmetry boundary conditions were usedto reduce the problem size.For this problem, we observe the global distribution of error and single pointhistories of the solution.

1847.2.1 Global Distribution of ErrorsUsing the second-order accurate schemes sl2dAc-O2 and ul2dAc-O2, an instan-taneous solution is shown in Figure 7.30, and the corresponding instantaneous error(pnum�pex) is shown in Figure 7.31. (Note that for small errors, this error is propor-tional to the total phase error, but not for larger values). The time correspondingto these �elds is long enough after the start of the integration so that the initialtransients have already subsided. As expected, the standard leapfrog scheme has thelargest error along the axes and the upwind leapfrog scheme has it away from theaxes.The maximum error plotted next in Figure 7.32 is de�ned as the maximum ofjpnum � pexj recorded over several periods after the initial transients have subsided.The eventual leveling out displayed in this measure of error is due to the total phaseerror (equation 2.87) reaching 12 where the numerical solution is out of phase with theexact solution. Each error contour represents a 0.005 increase in error. Maximumerrors for the fourth- and sixth-order standard leapfrog schemes presented in Figures7.33 and 7.34 show marked improvements. Maximum errors for the upwind leapfrogschemes are presented in Figures 7.35, 7.36 and 7.37.As noted previously, keeping an already coarse resolution constant does not showall the potential values of a high-order scheme. The worst error is only halved ingoing from scheme sl2dAc-O4 to sl2dAc-O6. This would be much improved for aslightly higher resolution.Figures 7.35, 7.36 and 7.37 show a lack of expected symmetry about the quadrantbisector. This is due to the error being taken from faces normal to the x-direction.Error taken from faces normal to the y-direction shows an equal and opposite distri-bution, re ecting overall symmetry.

1855

1015

2025

305 10 15 20 25 30

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

x

y(a) Standard leapfrog scheme0

510

1520

2530

5 10 15 20 25 30

-0.1

-0.05

0

0.05

0.1

0.15

0.2

x

y(b) Upwind leapfrog schemeFigure 7.30: Instantaneous solution of 2D radiating wire problem.

1865

1015

2025

305 10 15 20 25 30

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

x

y(a) Standard leapfrog scheme0

510

1520

2530

5 10 15 20 25 30

-0.04

-0.02

0

0.02

0.04

x

y(b) Upwind leapfrog schemeFigure 7.31: Instantaneous error of 2D radiating wire problem.

1875

1015

2025

305 10 15 20 25 30

0

0.025

0.05

0.075

0.1

x

yFigure 7.32: Maximum error of 2D radiating wire problem using scheme sl2dAc-O2.5

1015

2025

305 10 15 20 25 30

0

0.01

0.02

0.03

0.04

0.05

x

yFigure 7.33: Maximum error of 2D radiating wire problem using scheme sl2dAc-O4.

1885

1015

2025

305 10 15 20 25 30

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

x

yFigure 7.34: Maximum error of 2D radiating wire problem using scheme sl2dAc-O6.0

510

1520

2530

5 10 15 20 25 30

0

0.02

0.04

0.06

0.08

x

yFigure 7.35: Maximum error of 2D radiating wire problem using scheme ul2dAc-O2.

1890

510

1520

2530

5 10 15 20 25 30

0

0.01

0.02

0.03

0.04

x

yFigure 7.36: Maximum error of 2D radiating wire problem using scheme ul2dAc-O4-A.0

510

1520

2530

5 10 15 20 25 30

0

0.002

0.004

0.006

0.008

x

yFigure 7.37: Maximum error of 2D radiating wire problem using scheme ul2dAc-O4-B.

190The error predictions compare very well with this numerical experiment for theupwind leapfrog schemes. A decrease to half the error level is predicted from schemeul2dAc-O2 to scheme ul2dAc-O4-A, and a decrease to one-sixth the error level fromscheme ul2dAc-O4-A to scheme ul2dAc-O4-B is re ected in the numerical experi-ments. This correlation is not so strong for the standard leapfrog schemes. Thefour-fold improvement in error from sl2dAc-O2 to sl2dAc-O4 and the three-fold im-provement from sl2dAc-O4 and sl2dAc-O6, as predicted by the analysis, materializedas roughly two-fold improvements in the numerical experiments. This is attributedto the assumption of plane waves in the analyses and restates the caution in relyingcompletely on the analyses.7.2.2 Single-Point HistoriesWe present next the history record of pressure at two points, one near the axisdirection and one near the most skewed direction. The points are located approxi-mately three wavelengths from the source. This history shows the evolution of thetransient solution as well as the eventual development into the harmonic solution.These histories should indicate the relative reliability of the transient solutions incomparison to the harmonic solutions, for which the analyses were carried out inthe previous chapters. The numerical and exact solution histories are shown in Fig-ures 7.38, 7.39 and 7.40 for the 2D standard leapfrog schemes and in Figures 7.41,7.42 and 7.43 for the 2D upwind leapfrog schemes. Although both schemes workedwith the same spatial resolution, the temporal resolution is coarser for the standardleapfrog scheme because any variable is only available at every other time step.The standard leapfrog schemes tend to overpredict the amplitudes of the waves.Even the sixth-order accurate scheme does this. The high amplitude is not an insta-

191bility but it is a bit surprising since no amplitude error is expected for this familyof schemes. The overprediction may be due to the manner in which the source isincorporated into the numerical algorithm. On top of the amplitude overpredictionof the harmonic solutions are strong overshoots of the �rst one or two extrema forthe schemes of any order and a slow build-up to the harmonic solution amplitude forthe second-order scheme along the axis direction. Perhaps including time derivativesof the source terms may improve the amplitude predictions. One case in which theanalysis is right on the mark is that at the most skewed angle (and at the stabilitylimit) the phase error of the harmonic solution is (virtually) perfect, even for thelargely inaccurate second-order scheme.The upwind leapfrog scheme does not do as poorly with amplitude predictions,but again, it is surprising to see any amplitude error, these schemes are also from afamily of reversible schemes. The high-order upwind leapfrog schemes seem to haveproblems with the curvatures of the peaks and valleys of the signal along the axisdirection during the initial cycles. This may be due to the generation of spuriousmodes. One disadvantage of the upwind leapfrog scheme is that because more char-acteristic variables exist than primitive variables, more spurious modes are generatedfor these schemes than for the standard leapfrog schemes.7.3 Three-Dimensional Pressure SourceThe pressure source problem of the last section is examined here for the 3Dschemes. The numerical implementation of the source term is virtually identical tothat in 2D. In 3D, the exact solution for the pressure term ispex = �4�c cos h(�r + �2 )� (!� + ')ir (7.5)where again ' is set to ��2 to prevent a discontinuity at the start.

192(a) axis direction-0.1

-0.08-0.06-0.04-0.02

00.020.040.060.080.1

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(b) 2D skewed direction-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp Figure 7.38: Histories for 2D pressure source using scheme sl2dAc-O2.For each scheme, the point history is presented for three locations, near the axisdirection, near the 2D skewed direction and near the 3D skewed direction. The his-tories are presented in Figures 7.44, 7.45 and 7.46 for the standard leapfrog schemesand Figures 7.47, 7.48 and 7.49 for the upwind leapfrog schemes.The 3D standard leapfrog schemes behave very much like their 2D versions. Theyshow amplitude overpredictions that persist through the sixth-order accurate scheme.The overshoots for the �rst one or two extrema are very large in the two skeweddirections, even doubling the amplitude of the exact solution. The second-orderaccurate version exhibits a slow build-up of the signal along the axis direction, as in2D, taking two cycles to reach the amplitude of the harmonic solution.The ul3dAc-O2 scheme overpredicts amplitude along the axis direction and over-predicts it along the 3D skewed direction, but this is much improved in the two

193(a) axis direction-0.1

-0.08-0.06-0.04-0.02

00.020.040.060.080.1

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(b) 2D skewed direction-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp Figure 7.39: Histories for 2D pressure source using scheme sl2dAc-O4.fourth-order versions. Very fast errors are very apparent for the upwind leapfrogschemes along the axes due to the stability limit of 13, rather than 12 , which wouldhave given much more accurate results in this direction. Other than this highlyvisible error, the upwind leapfrog schemes do quite well.7.4 Three-Dimensional Dipole SourceThree-dimensional results for the problem of a dipole is presented in this section.A z-directed electric dipole withJ e;z(� ) = cos (!� + ') (7.6)is located at coordinates (0; 0; 0) and ' = ��2 . The exact solution for Ez isEz = �4�r "(1� z2) + 2(�r)2 z2+# cos�(�r + �2 )� (!� + ')�

194(a) axis direction-0.1

-0.08-0.06-0.04-0.02

00.020.040.060.080.1

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(b) 2D skewed direction-0.1-0.08-0.06-0.04-0.02

00.020.040.060.080.1

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp Figure 7.40: Histories for 2D pressure source using scheme sl2dAc-O6.� �4�r 2�r z sin�(�r + �2 )� (!� + ')� (7.7)where z = zpx2+y2+z2 . The presence of the dipole modi�es the standard leapfrogdi�erence equation 4.8a toEzn+1;i;j;k � Ezn+1;i;j;k2�� + c ��yn;i;j;kHx ��xn;i;j;kHy�+ J e;z(n�� ) = 0 (7.8)and the upwind leapfrog di�erence equation 4.9a toGxyn+1;i+1=2;j;k �Gxyn;i+1=2;j;k +Gxyn;i�1=2;j;k �Gxyn�1;i�1=2;j;k2�� + ci;j;k�xn;i;j;kGxy�ci;j;k2 h�yn;i;j;k(Gyxy + Syxy ) + �zn;i;j;k(�Gzyz + Szyz )i+ J e;z(n�� )2 = 0 (7.9)and other equations follow similarly.The histories are presented in Figures 7.50, 7.51 and 7.52 for the standard leapfrogschemes and Figure 7.53 for the upwind leapfrog schemes.

195(a) axis direction-0.08

-0.06-0.04-0.02

00.020.040.060.080.1

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(b) 2D skewed direction-0.08-0.06-0.04-0.02

00.020.040.060.08

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp Figure 7.41: Histories for 2D pressure source using scheme ul2dAc-O2.A clear pattern has persisted in all the standard leapfrog schemes. They showamplitude overpredictions through even the highest-order scheme and in the earlytransient cycles, large overshoots away from the axis direction. The second-orderscheme shows the same slow build-up along the axis-direction, taking two cyclesto reach the amplitude of the harmonic solution. Phase error remains excellent inthe most skewed direction and improves with the order of accuracy in the otherdirections.Unlike the 3D acoustic problem, the upwind leapfrog scheme does not exhibitstrong fast errors along the axis direction because it is highly accurate in that di-rection. It does, however, exhibit a slow build-up of amplitude along the 3D skeweddirection, (but not to the degree of the second-order accurate standard leapfrogschemes) taking one cycle to reach the amplitude of the harmonic solution. It ex-

196(a) axis direction-0.08

-0.06-0.04-0.02

00.020.040.060.080.1

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(b) 2D skewed direction-0.08-0.06-0.04-0.02

00.020.040.060.08

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp Figure 7.42: Histories for 2D pressure source using scheme ul2dAc-O4-A.hibits strong overshoots of the �rst or second extrema, more noticeably than it didfor the acoustic problem. The severe overshoots also appear more prominently inthe directions of least accuracy when they are present.The 3D standard leapfrog schemes for electromagnetics behave very similarly tothe 3D standard leapfrog schemes for acoustics, as the analysis predicted. In the in-terest of �nding if such similarity exists for the upwind leapfrog schemes, Figure 7.54presents the second-order accurate scheme's results using � = 13 , the Courant num-ber for the acoustic problem. There are strong, very high-frequency errors movingahead of the transient phase along the axis direction. The overshoots in the initialcycles of the transients have disappeared. The most accurate history is predictedalong the 2D skewed direction. There is still a slight amplitude overprediction inthe harmonic solution along the axis and a more signi�cant undershoot along the 3D

197(a) axis direction-0.08

-0.06-0.04-0.02

00.020.040.060.080.1

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(b) 2D skewed direction-0.08-0.06-0.04-0.02

00.020.040.060.080.1

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp Figure 7.43: Histories for 2D pressure source using scheme ul2dAc-O4-B.skewed direction. These features are similar to those exhibited by the 3D upwindleapfrog scheme for acoustics at � = 13 .

198(a) axis direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(b) 2D skewed direction-0.02-0.015-0.01

-0.0050

0.0050.01

0.0150.02

0.025

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(c) 3D skewed direction-0.03

-0.02

-0.01

0

0.01

0.02

0.03

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp Figure 7.44: Histories for 3D pressure source using scheme sl3dAc-O2.

199(a) axis direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(b) 2D skewed direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(c) 3D skewed direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp Figure 7.45: Histories for 3D pressure source using scheme sl3dAc-O4.

200(a) axis direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(b) 2D skewed direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(c) 3D skewed direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp Figure 7.46: Histories for 3D pressure source using scheme sl3dAc-O6.

201(a) axis direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(b) 2D skewed direction-0.01-0.008-0.006-0.004-0.002

00.0020.0040.0060.0080.01

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(c) 3D skewed direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp Figure 7.47: Histories for 3D pressure source using scheme ul3dAc-O2.

202(a) axis direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(b) 2D skewed direction-0.015

-0.01

-0.005

0

0.005

0.01

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(c) 3D skewed direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp Figure 7.48: Histories for 3D pressure source using scheme ul3dAc-O4-A.

203(a) axis direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(b) 2D skewed direction-0.01-0.008-0.006-0.004-0.002

00.0020.0040.0060.0080.01

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp

(c) 3D skewed direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cp Figure 7.49: Histories for 3D pressure source using scheme ul3dAc-O4-B.

204(a) axis direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cEz

(b) 2D skewed direction-0.025-0.02

-0.015-0.01

-0.0050

0.0050.01

0.0150.02

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cEz

(c) 3D skewed direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cEz

Figure 7.50: Histories for 3D dipole source using scheme sl3dEm-O2.

205(a) axis direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cEz

(b) 2D skewed direction-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cEz

(c) 3D skewed direction-0.01

-0.005

0

0.005

0.01

0.015

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cEz

Figure 7.51: Histories for 3D dipole source using scheme sl3dEm-O4.

206(a) axis direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cEz

(b) 2D skewed direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cEz

(c) 3D skewed direction-0.01

-0.005

0

0.005

0.01

0.015

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cEz

Figure 7.52: Histories for 3D dipole source using scheme sl3dEm-O6.

207(a) axis direction-0.02

-0.015-0.01

-0.0050

0.0050.01

0.0150.02

0.025

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cEz

(b) 2D skewed direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cEz

(c) 3D skewed direction-0.008-0.006-0.004-0.002

00.0020.0040.0060.0080.01

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cEz

Figure 7.53: Histories for 3D dipole source using scheme ul3dEm-O2.

208(a) axis direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cEz

(b) 2D skewed direction-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cEz

(c) 3D skewed direction-0.008-0.006-0.004-0.002

00.0020.0040.0060.008

-10 -5 0 5 10 15 20 25 30 35

exactnumerical� � r=cEz

Figure 7.54: Histories for 3D dipole source using scheme ul3dEm-O2 at lower � = 13.

CHAPTER VIIIFINAL WORDSIn this chapter, we summarize our �ndings, draw some conclusions from themand o�er some �nal words.8.1 Summary and ConclusionsThis thesis considered some approaches to more accurate and e�cient time-domain methods for simulating the scatter of electromagnetic and acoustic waves.We concentrated on the class of three-level time-domain schemes which have receivedless attention to two-level schemes.We began with the characteristics analyses for the two systems of equations foracoustics and electromagnetics. Introducing the bicharacteristic forms of these sys-tems of equations, we showed how wave interactions can be written in terms of char-acteristic variables and devised an extension for the successful 1D upwind leapfrog�nite di�erence scheme of Iserles to be used in these multi-dimensional problems.We began the numerical schemes in 1D with a multitude of schemes for linear ad-vection categorized by two discriminations: upstream-biased versus central-di�erenceand di�usive versus non-di�usive. The upstream-biased schemes in general have bet-ter phase errors under the conditions for each individual scheme to perform its best,209

210but the upstream-biased schemes degrade more rapidly than the central-di�erenceschemes do. For small Courant numbers, the upstream-biased schemes are less ac-curate than the central-di�erence schemes. The upwind leapfrog schemes had twicethe stability range of the central-di�erence schemes (but part of this superiority islost in higher dimensions). There are notable advantages and disadvantages to boththe di�usive and the reversible non-di�usive schemes. There is more exibility in thedesign of di�usive schemes, while non-di�usive schemes must be reversible. Capital-izing on this exibility, an optimizable scheme adjustable for better phase error orbetter amplitude error was introduced. Di�usive schemes damp out high-frequencywaves faster. In wide-band problems, where there exist extreme frequencies that maybe disregarded, di�usive schemes destroy these frequencies before they can in uencethe progression of solution. This is a very nice feature for non-linear problems inwhich a small error can have a large e�ect. But there are admittedly practical prob-lems where phase errors are not as important and much longer computation timescan be carried out by the non-di�usive schemes, as the di�usive schemes eventuallyzero out the solution. Many of these features and di�erences were brought out bythe numerical experiments in Chapter VII. Di�usive schemes seem to be better incontrolling the high-frequency errors that exist in wide-band excitations and main-taining the shape of a pulse disturbance, but for harmonic problems, their inabilityto maintain the strength of the disturbance for long integration times leaves them ata disadvantage compared with the non-di�usive schemes. In all cases, errors can becontrolled by resolution, and the faster convergence of the high-order schemes makethem more ideal when error tolerances are small.From the 1D schemes, we developed the non-di�usive multi-dimensional upwindleapfrog schemes for acoustics and electromagnetics and compared them to Yee's

211standard leapfrog scheme and to our many generalizations for that scheme. As in1D, the upwind leapfrog schemes did better at the stability limit but degraded morerapidly than the central-di�erence schemes away from this limit. All schemes hadreduced stability limits as the number of dimensions increased, but interestinglythe upwind leapfrog scheme for electromagnetics did not. In 3D, its stability limitremains � = 12, same as in 2D. This is a great advantage because the upwind leapfrogschemes are more accurate for higher Courant numbers and at � = 12 , several termsin the update equation drop out and do not have to be computed.The high-order extensions for the standard leapfrog schemes were straight for-ward. The extensions for the upwind leapfrog schemes presented several schemesthat di�er by the stencil on which some multi-dimensional derivatives were resolved.The choice of which to use made a signi�cant di�erence in the numerical error. In thecase of 3D acoustics, the worst errors were not a�ected much (if any) by this choice.In 2D, the e�ects were very signi�cant in both the analysis and in the numericalexperiments. The best choice had not only one-seventh the error of the worst choice,but it also required the least amount of computations of all the choices. When anupwind leapfrog scheme is used with � = 12 , the scheme can be even further simpli-�ed. The upwind leapfrog scheme for electromagnetics showed great promise by nothaving a decrease in stability limit in going from 2D to 3D. Unfortunately, no stablefourth-order accurate extension was found for this scheme, hence no fourth-orderupwind leapfrog scheme was able to take advantage of the special Courant numberof 12 in 3D.For high-order schemes, analyses shows clearly the gain in accuracy at a �xedresolution. In addition, the rate at which the error drops with respect to resolutionimproved signi�cantly. Increases in order of accuracy should not be relied upon alone

212to improve accuracy. Rather, there should be a combination of increases in order ofaccuracy and resolution (especially for an already-low resolution) to bring about themost signi�cant improvements to error.Our study of the e�ciency (or cost-e�ectiveness) of the standard leapfrog and up-wind leapfrog schemes in multi-dimensions showed how the cost of the more complexschemes balance out with their higher accuracy. Memory requirements are alwaysdecreased when switching from a lower-order scheme to a higher-order scheme butnot op requirements. In general, at a \reasonable" error requirement, say 1%, up-grading from a second-order scheme to a fourth-order scheme saves computations.But upgrading from a fourth-order scheme to a sixth-order scheme does not savecomputations until the error requirement is about an order of magnitude smaller.For both acoustic and electromagnetic problems in 2D and 3D, the fourth-orderaccurate standard leapfrog and upwind leapfrog schemes are signi�cant improvementsover the widely-used Yee's second-order electromagnetic standard leapfrog schemeand its analog for acoustics. In 2D, the fourth-order upwind leapfrog scheme wasover the most e�cient in both measures of e�ciency, except when compared to themore memory-e�cient sixth-order accurate standard leapfrog scheme. For 3D, anupwind leapfrog scheme always uses more memory than a standard leapfrog schemeof the same order of accuracy. The computational operations count are more evenbetween standard leapfrog and upwind leapfrog schemes.Because the e�ciency analysis was performed only for the bare essentials of thescheme, it is important to remember that the outcome of the comparison may changefor slightly di�erent implementations. For example, inhomogeneous media problemswould favor the more complex schemes, and boundary conditions are easier with thelower-order accurate schemes.

213Numerical experiments presented in Chapter VII for the 1D propagation of a(wide-band) pulse, the di�usive schemes did a better job of maintaining the shapeof the pulse. This was attributed to the ability of these schemes to damp out high-frequency components that had the worst phase error. For the propagation of acontinuous wave, the di�usive schemes showed their greatest weaknesses in grossamplitude errors, while the non-di�usive schemesmaintained all the wave amplitudes.The numerical experiments showed that the tunable scheme was e�ective in its abilityto be adjusted as needed.The multi-dimensional experiments examined the distribution of the errors (in2D), the evolution of transients due to the input of source terms into the numericalschemes. The error distribution in 2D agreed to a great extent with the analyses,which is not trivial because the analyses assumed planar waves and the problemsinvolved curved wave fronts. In the directions for which the analyses predicted perfectphase speed, the actual numerical results still showed �nite error. This may be dueto the curvature of the wave front or imperfect treatment of the source terms. Thepoint histories of the solutions revealed some persistent amplitude overpredictions bythe standard leapfrog schemes for both the acoustic and electromagnetic problems.The high-order standard leapfrog schemes had less error, but the amplitude errorswere consistently greater than the phase errors. Because these schemes are expectedto have no amplitude error, the error may be due to the implementation of the source.Amplitude errors appeared to a lesser degree for the upwind leapfrog schemes andare su�ciently correctable with the high-order schemes. In the transient phase, thestandard leapfrog scheme tended to either overshoot the �rst two extrema or buildup too slowly, compared with the amplitude of the eventual harmonic solution. Theupwind leapfrog schemes did much better when run at � = 12 , but for � = 13 , which

214is the limit of the 3D acoustic scheme, very high frequency errors tend to shoot outahead of the pack along the axis direction. These problems shown by the numericalexperiments stress that while the interior di�erence scheme may be very accurate,some attention needs to be directed to other sources of error, such as source termsand probably boundary conditions.8.2 Further WorkThe new upwind leapfrog schemes presented in this dissertation have severaluseful properties that make them desirable. They are signi�cantly less troublesomenear boundaries and across surface discontinuities. They are more exible towardpossible extensions for exotic material and non-linearity. They can use coarser grids,which is a saving when additional data (other than the solution) is stored on thegrid. The upwind leapfrog grid storage scheme partition more naturally in parallelimplementations. Most of the upwind leapfrog schemes presented in this dissertationare not more e�cient than the high-order accurate extensions of Yee's scheme. Whencombined with their other desirable properties and considering speci�c problems,they may be more appropriate. Some additional work to expand the usefulness ofthe upwind leapfrog schemes and extend the ideas and techniques presented in thisdissertation are given in this section.Hybrid second-fourth-sixth order schemes have been suggested in this dissertationto gain the accuracy of high-order schemes while retaining the relative ease of thesecond-order scheme in dealing with boundaries. Such schemes would use the mostaccurate update method possible but drop to a lower method at the boundaries. Thefeasibility of such hybrid schemes should be examined. To make up for the loss ofaccuracy at the boundaries, there could be a halving or quartering of the grid spacing

215there. Such a scheme is formally second order, but still can be very accurate, sincelow-accuracy computation is performed in a space that is one spatial degree smallerand does not have as much room to grow. The upwind leapfrog scheme will be agood candidate for this approach because it is most accurate along grid directions,where the low-order scheme must be used.A strong motivation exists to determine exactly why the higher-order 3D upwindleapfrog schemes for electromagnetics are unstable and to seek a stable version.Because the 3D second-order upwind leapfrog scheme for electromagnetics is capableof running at � = 12, where it is optimal, it is regrettable that a high-order extension isnot yet found. One approach to a stabilizing modi�cation is to note that the presenceof the vorticity discretization seemed to a�ect the stability of the 2D fourth-orderupwind leapfrog scheme at � = 12 , even when vorticity should be zero. Perhaps theaddition of a divergence term on the order of the truncation, not a�ecting consistencyof the equation, will stabilize the 3D schemes that are currently unstable.The initial e�orts at extending the upwind leapfrog scheme to higher dimensions�rst dealt with the inclusion of the cross terms into the scheme. Since the updateequations outnumber the primitive variables, the staggering method used in thiswork was created so that averaging updates can be avoided, and the non-di�usivenature of the 1D scheme can be retained. In light of the 1D analyses and results, itmay be fruitful to give the di�usive schemes another look, with two possible goals.The �rst is to �nd an averaging algorithm which does more good than harm. Suchan averaging algorithm may, like the optimized scheme of Section 3.7, di�use thespurious modes faster than the physical mode. Or it may di�use the modes travelingat the incorrect speeds more than it does the ones traveling at more accurate speeds.The �nal argument for averaging is that a more standard storage scheme may be

216used, which lowers the memory requirement and places all variables at the samepoint to allow more extensions for physical complex interactions between the �elds.We have mentioned that the stencils on which error terms were resolved are notunique but rather chosen for their compactness in discretizing each individual term.It is possible to choose less compact stencils to obtain slightly di�erent schemes. The�rst reason to attempt this is that the union of all stencils used to update a variablemay not be less compact, even when some of the individual stencils comprising itare less compact. The second reason is that the error of the scheme will change andmay improve.

APPENDICES

217

218APPENDIX AEIGENSYSTEM OF THEELECTROMAGNETIC JACOBIAN MATRIXThis appendix describes a characteristic analysis procedure that is well knownand often used in CFD. Here, it is used to derive the electromagnetic characteristicvariable vector G.A.1 Eigensystem AnalysisThe Jacobian matrix (2.33) for Maxwell's equation has the eigenvalues� = 0; 0; c; c;�c;�c (A.1)where c = 1=p�� is the speed of light.The right and left eigenvectors for the two stationary waves, associated with � = 0and called the E0 and H0 waves, are simple and similar to each other:rE0 = lH0 = 0BBBBBBBBBBBBBBBBBBBB@ 000nxnynz1CCCCCCCCCCCCCCCCCCCCA (A.2)

219rH0 = lE0 = 0BBBBBBBBBBBBBBBBBBBB@ nxnynz0001CCCCCCCCCCCCCCCCCCCCA (A.3)Note that the right-left eigenvector pairs are orthonormal for both of the stationarywaves. Since the eigenvalue is repeated, any linear combination of rE0 and rH0 isalso a right eigenvector, and any linear combination of lE0 and lH0 is also a lefteigenvector. The choices (A.2, A.3) were made to lead to the separation of electricand magnetic �elds in the wave strengths (2.35, 2.36) and to the explicit conclusionthat electromagnetic waves are transverse waves.The remaining eigenvalues belong to four moving waves; two at speed +c andtwo at speed �c along bn.The eigenvectors corresponding to the moving waves are more di�cult to ex-press. Since each eigenvalue is repeated, each of their eigenvectors can lie in atwo-dimensional subspace of the six-dimensional space of the dependent variables.This leads to a certain degree of ambiguity in writing the eigenvectors, because thereare two linearly independent choices but a three-way symmetry due to the three-dimensional �eld vectors. Some \natural" ways of writing the right eigenvectors (r)

220and left eigenvectors (l) for the +c waves can be given byrE;x = 12 0BBBBBBBBBBBBBBBBBBBB@ 1 � n2x�nxny�nxnz0Znz�Zny1CCCCCCCCCCCCCCCCCCCCA ; rE;y = 12 0BBBBBBBBBBBBBBBBBBBB@ �nxny1� n2y�nynz�Znz0Znx

1CCCCCCCCCCCCCCCCCCCCA ; rE;z = 12 0BBBBBBBBBBBBBBBBBBBB@ �nxnz�nynz1 � n2zZny�Znx01CCCCCCCCCCCCCCCCCCCCA (A.4)

lE;x = 12 0BBBBBBBBBBBBBBBBBBBB@ 1 � n2x�nxny�nxnz0Y nz�Y ny1CCCCCCCCCCCCCCCCCCCCA ; lE;y = 12 0BBBBBBBBBBBBBBBBBBBB@ �nxny1� n2y�nynz�Y nz0Y nx

1CCCCCCCCCCCCCCCCCCCCA ; lE;z = 12 0BBBBBBBBBBBBBBBBBBBB@ �nxnz�nynz1� n2zY ny�Y nx01CCCCCCCCCCCCCCCCCCCCA (A.5)(The factor 12 is arbitrary, since the eigenvectors can only be speci�ed to an arbitraryconstant factor. We choose 12 by our convention.) Note that three di�erent left andthree di�erent right eigenvectors are described for each wave, but only two are linearlyindependent for each left or right eigenvectors. (As seen below in Section A.2.1, stillthree more \natural" ways can be derived for each left or right eigenvector, but thenumber of linearly independent eigenvectors remain two.)A.2 Characteristic VariablesDespite the unavoidable ambiguity in the choices of eigenvectors, it is possibleto uniquely resolve electromagnetic disturbances into waves by considering innerproducts of any left eigenvector with Q.

221The characteristic variable associated with the moving waves are determined bythe inner product of the above left eigenvectors. Since only two can be independent,the choice of eigenvectors must be made carefully to obtain a concise de�nition forthe characteristic variable vector. The de�nitions above do lead to fairly concise andintuitively meaningful characteristic variables.We premultiplyQ with the left eigenvectors (A.5), obtaining the wave strengthsGnx = lE;x �Q = (1 � n2x)Ex � nx(nyEy + nzEz)� (nyHz � nzHy)2 (A.6a)Gny = lE;y �Q = (1 � n2y)Ey � ny(nxEx + nzEz)� (�nxHz + nzHx)2 (A.6b)Gnz = lE;z �Q = (1 � n2z)Ez � nz(nxEx + nyEy)� (nxHy � nyHx)2 (A.6c)where the �rst superscript on G speci�es the propagation direction associated withthe characteristic variable and the second speci�es the component that forms the\characteristic variable vector" G+n = Et � n�H2 (A.7)The superscript t on E stands for the transverse (to bn) components. By de�nition,G+n is transverse to bn, and although it is a three-dimensional vector, it has only twoindependent components. This is consistent with the fact that only two of the eigen-vectors used in its de�nition are independent, although we used three eigenvectorsin (A.6).The single characteristic variable vector then represents both of the waves corre-sponding to � = +c (traveling with speed c along +bn). For the waves correspondingto � = �c (traveling with speed c along �bn), one needs only to reverse the sign of bnin the analysis.

222A.2.1 Alternate De�nitionsA set of eigenvectors similar to (A.4, A.5) can be added to the list of \natural"forms of the eigenvectors:rH;x = 0BBBBBBBBBBBBBBBBBBBB@ 0�Y nzY ny1� n2x�nxny�nxnz1CCCCCCCCCCCCCCCCCCCCA ; rH;y = 0BBBBBBBBBBBBBBBBBBBB@ Y nz0�Y nx�nxny1� n2y�nynz

1CCCCCCCCCCCCCCCCCCCCA ; rH;z = 0BBBBBBBBBBBBBBBBBBBB@ �Y nyY nx0�nxnz�nynz1� n2z1CCCCCCCCCCCCCCCCCCCCA (A.8)

2lH;x = 0BBBBBBBBBBBBBBBBBBBB@ 0�ZnzZny1 � n2x�nxny�nxnz1CCCCCCCCCCCCCCCCCCCCA ; 2lH;y = 0BBBBBBBBBBBBBBBBBBBB@ Znz0�Znx�nxny1 � n2y�nynz

1CCCCCCCCCCCCCCCCCCCCA ; 2lH;z = 0BBBBBBBBBBBBBBBBBBBB@ �ZnyZnx0�nxnz�nynz1� n2z1CCCCCCCCCCCCCCCCCCCCA (A.9)They are obtained from linear combinations of (A.4, A.5) for the same eigenvalue,so they are also eigenvectors.Working with these lH eigenvectors rather then the lE eigenvectors would yielddi�erent wave strengths and the characteristic variable vectorG+n� = Ht + Y n�E2 (A.10)but the results, in terms of the vectors E and H (Section 2.5) would be the same.Additionally, one can show n�G�n� = �YG�n (A.11)

223APPENDIX BEQUIVALENCE OF YEE'S AND STANDARDLEAPFROG SCHEMESIn this Appendix, we note the similarity between the standard (central-di�erence)leapfrog scheme and a generalization of Yee's scheme. As shown in many text books,e.g., [57], the standard leapfrog scheme is a simple second-order accurate, three-levelcentral-di�erence scheme for �rst-order di�erential equations. The point-centered(centered around a point on the computational grid), three-level, central-di�erenceapproximation of the time derivative does not involve the central point of the stencil.The approximations of �rst-order spatial derivatives also do not involve this point.Each update replaces the solution at the time step (n + 1) with the solution attime step (n � 1), \leapfrogging" over the solution at time step n. Yee describedthe second-order accurate �nite-di�erence time-domain scheme for the 3D Maxwell'sequation as a central-di�erence scheme with a grid that is staggered in space andtime [14]. The original Yee's scheme uses a rectangular grid, placing the tangentialcomponent of E at the line midpoints and placing the normal component of H atthe face centers, as shown in Figure B.1c. Hence the components reside on grids thatare staggered with respect to each other. Staggering in time is done by keeping theE �eld components at integer time steps and the H �elds at half-integer time steps.

224Time step size is considered to be from one integer time step to the next integer timestep.We will show that Yee's scheme is a subset of the standard leapfrog scheme.Consider the standard leapfrog scheme on a grid twice as �ne as Yee's grid. Thestandard leapfrog scheme is a more general and \naive" scheme in which all variablesare stored at all grid points (Figure B.1a). The time step size is from one time levelto the very next time level rather than over two time levels as in Yee's scheme. Upondiscretizing Maxwell's equations, some interesting decoupling of the discrete solutionoccurs. First, the E �elds on one time level decouple from the E �elds on adjacenttime levels; so do the H �elds. Figure B.1b shows the data storage of one of the twodecoupled sets. The E and H �elds in this set interact with each other, but theyreside on adjacent time levels rather than the same time level. At each update, one�eld leapfrogs over the other in alternating fashion.Next, within the decoupled set of Figure B.1b, the components again decouple,this time, into four sets, as shown in Figures B.1c-f. Each E and H vector in FigureB.1b distributes its x-, y- or z-component, or nothing, to each of the four new sets.Figure B.1c shows the set corresponding to Yee's scheme. Note that each set containsall the variables in locations appropriate for applying the standard leapfrog scheme,but one set does not require data from any of the other sets. Note also that there isno special property of Yee's set, since a simple translation of coordinates can changethe other sets to Yee's set. Four more sets exist, identical to these except that theE and H �elds are switched, corresponding to the second set of the �rst decoupling.Although Yee's scheme yields the same solution for its mode as would the stan-dard leapfrog scheme (on a doubly �ne grid), it is a much more e�cient scheme.Clearly, less work is required since only one decoupled set is updated and the rest ig-

225E,HE,HE,HE,H E,H E,H E,H E,HE,HE,HE,H E,HE,HE,HE,HE,HE,H E,HE,HE,Ha. E and H �elds stored at grid pointsdenoted by circles. HHH E EH HHEH E E H E HEH EHH HEEH E Eb. E and H �elds decouple. E residesin every other time level and H re-sides in the remaining time levels.E E E EEEE EE EH HHH HHc. Components decouple. This isYee's scheme. Bold lines show Yee'sgrid. HE E EE

E EEE EH H HHH HH H HHd. Alternate decoupling.E HH HHHH H H HH HE EEE E EE Ee. Alternate decoupling. HE E E EE E E EHH H HHH HHf. Alternate decoupling.Figure B.1: Decoupling of the standard leapfrog scheme into Yee's scheme. Thicklines denote Yee's grid. Thin lines denotes the doubly �ne standardleapfrog grid.

226nored. As shown, Yee's scheme requires one-eighth the work of the standard leapfrogscheme (while attaining the same solution).Less memory is also required by Yee's scheme. With respect to the �ne grid, thestandard leapfrog scheme of Figure B.1a stores six oating points per cell and muststore two time levels of solutions. (The third time level can over-write the �rst.)The scheme of Figure B.1b stores three-halves of a oating point per cell for the E�eld plus at an adjacent time level, three-halves of a oating point per cell for theH �eld, for a total of three oating points per cell. Any of the schemes in FiguresB.1c-f would requires one-fourth this amount, or three-fourths of a oating point percell. Thus Yee's scheme uses one-eighth of the memory requirement of the standardleapfrog scheme.The 2D Yee's scheme can be found by assuming zero gradients in the z-direction.The TM scheme is on any x-y plane containing a location where an Ez is kept. TheTE scheme is the same, but goes through a location where Hz is kept. The argumentdescribing Yee's scheme as a subset of the standard leapfrog scheme is analogous in2D. Although reference [14] did not describe a 1D version of the scheme, such ascheme is simply that on a grid line going through locations where any componentof E is located and locations where any component of H is located.Yee's scheme is signi�cantly less costly than the standard leapfrog scheme whilegiving the same solution. It does not give all the solutions of the standard leapfrogscheme, but the remaining solutions can be interpolated from the Yee set, incurringan additional one-time error over the error that accumulates each time step.There are examples in which the standard leapfrog scheme does not decouple asshown above, and these lead to the limits of Yee's scheme. One such example is whenthe E and H �elds interact at the same point, such as boundaries and sources. The

227simple impedance boundary condition,bb�E = � � (bb�H) (B.1)where bb is normal to the boundary, is often used to relieve the expense of simulatinga conducing boundary under a thin dielectric coating. It requires transverse compo-nents of E and H at the same point. As clearly seen in Figure B.1c, this cannot beeasily applied with Yee's scheme, because the required data do not exist at the samepoint in space or in time. Schemes storing transverse-�eld components on grid faces,such as the upwind leapfrog schemes easily accomplishes this. Another example offailure to decouple is for more general versions of Maxwell's equations. In chiralmedia for which the constituent relationships are~D = "~E � � ~H� (B.2a)~B = � ~H + �~E� (B.2b)the time discretization involves the previously decoupled point at the center of thestencil, causing a failure of the standard leapfrog scheme to decouple. (Indeed, onecan say it is no longer a leapfrog scheme at all.) Media with general conductiv-ity matrices or general anisotropic constituent relationships also cause a failure todecouple.Although reference [14] did not describe the scheme for acoustics, the logicalextension of this scheme to acoustics is also one of eight decoupled sets of a standardleapfrog scheme on a doubly �ne mesh. This scheme has analogous advantages anddrawbacks.Now that Yee's scheme is shown to be a subset of the standard leapfrog scheme,it remains to resolve a matter of con icting de�nitions. Recall that in [14], the time

228step is de�ned between consecutive levels where the E �elds are de�ned, rather thanbetween any two consecutive levels. That is half as large as those de�ned in thisdissertation. We choose to de�ne the time step size as that between any adjacenttime steps to clarify comparisons with the upwind leapfrog schemes. Hence, thestability limits reported for Yee's scheme in this dissertation will be half as large,numerically, as that reported in [14] and other works.

229APPENDIX CMETHOD OF ANALYSISC.1 Classical von Neumann AnalysisThe von Neumann analysis [51] is extensively used to determine the stabilityand accuracy of the schemes in this dissertation. This method assumes that generalsolutions to the discrete equations can be written as a sum of planar harmonicsolutions (modes) having the formexp I(�xx+ �yy + �zz � c�� ) (C.1)where �x, �y and �z are the wave numbers for each of the coordinate directions and� = q�2x + �2y + �2z (C.2)is the total wave number. This harmonic assumption holds for linear equationsincluding those examined in this dissertation.On a uniform Cartesian grid, with xi;j;k = ih, yi;j;k = jh, zi;j;k = kh and � = n�� ,the form C.1 leads to the general form of the solutionQn;i;j;k = Q0 exp I(i�x + j�y + k�z � n�) (C.3)with � = �� (C.4)

230and �h = � = q�2x + �2y + �2z (C.5)For any harmonic mode, the discrete algebraic equations for a point in the gridcan be written in the form Qn+1 = MQn (C.6)or Qn = (M )nQ0 (C.7)where Qn is the vector of unknowns at time step n, at any point in the grid, and Mis the matrix of coe�cients of the discrete equations. The matrix M is also referredto as the complex ampli�cation matrix, and its eigenvalues are called the complexampli�cation factors for each eigen mode of M . The entries of M are functions of�x, �y, �z and the Courant number �.If we project Q0 onto the L right eigenvectors of M ,Q0 = LXl=1 �lrl (C.8)then we can write (C.7) as a sum of independent eigen modesQn = LXl=1 �l(M )nrl = LXl=1 �l(�l)nrl = LXl=1 �lj�ljneIn�lrl (C.9)with �l = j�ljeI�l (C.10)each of which can be examined individually.A grid will admit discrete solutions (exact or with error) having�x; �y; �z 2 [0; �] (C.11)

231At the upper limit (saw-tooth mode) the highest frequency admitted correspondsto �x = �y = �z = � (unless of course, one or more are equal to zero in lowerdimensional problems). The highest frequency is equivalent to a wavelength of 2hpd ind-dimensions. (Although such high frequencies would never be intentionally excited,they can be easily introduced by round-o� errors, and un-smooth wave forms andgeometry and boundary conditions.) Clearly a scheme is stable for solutions of theform C.3 if the spectral radius of M does not exceed unity for any admissible phase-angle combinations (C.11).Comparing C.9 with C.1 reveals that exact temporal phase angle should be�ex = ��� (C.12)The l-th mode propagates with temporal phase angle �l. Of the L modes, someare physical and some are spurious, some are stationary (arg �l = 0) and some arepropagating (arg �l 6= 0). We are usually concerned with the physical modes only(and the other modes to the extent that they must be stable). The physical modesare those whose right eigenvectors converge to corresponding eigenvectors in thephysical system. The convergence rate is the same as the order of accuracy of thediscrete equations. The propagation speed of the physical mode should also convergeto the exact propagation speed. For the determination of phase speed accuracy, let�num be the temporal phase angle associated with the physical mode and cnum be theassociated propagation speed. Then we have�num = ���(1 + �p) (C.13)or noting that propagation speed is proportional to the temporal phase angle,cnum = cex��num�� (C.14)

232C.2 Numerical Computations of EigensystemsFor small systems such as the second-order accurate, 1D problems of ChapterIII, the von Neumann analysis can be carried out almost completely symbolically,whether by hand or with the assistance of software (such as Mathematica or Maple)or both. For larger systems, such as most of the ones presented in this dissertation,this is virtually impossible. Instead of exact symbolic analysis, discrete numericalquantities are used in the computations. This leads to a certain degree of uncertaintyin the analysis, because only a �nite number of points in the parameter space canbe examined.The �rst uncertainty is the identi�cation of the physical mode of a numericalsolution. As mentioned above, strictly, this should be found by identifying the righteigenvector that converges to the exact eigenvector of the problem. This is di�cultsince the eigenvectors are functions of many parameters. The eigenvalues are alsofunctions of the same parameters, but they are a single number instead of L numbers.Rather than making many eigenvector computations to demonstrate convergence tothe exact eigenvector for a speci�c set of parameters, the eigenvalue that is closestto the exact eigenvalue is presumed to be that of the physical mode. Obviously, itis possible that the incorrect eigenvalue can be chosen. But in all cases presentedin this dissertation, the eigenvalue chosen is well behaved and convergent, giving aposteriori assurance that the choices made were correct.The second uncertainty is brought upon because instead of checking for stabilityfor the range in equation C.11, only a �nite number of points in the parameter spacecan be checked. In this dissertation, the scheme is presumed to be stable with thestability limit �max (de�ned for each individual scheme) if the numerical eigensystem

233analysis of its ampli�cation matrix yields stability for all combinations of� 2 �13�max; 23�max; �max� (C.15)and �x; �y; �z 2 ��; �2 ; �4 ; �8 ; �16� (C.16)

234APPENDIX DDISCRETE FINITE DIFFERENCEOPERATORSThe de�nitions of the discrete �nite di�erence operators � are de�ned here. Theyare based on simple second-order accurate central-di�erences, assuming a uniformCartesian grid with spacing h. The de�nitions are not repeated for symmetricallysimilar operators. Rather, additional operators can be derived by recalling that thex-, y- and z- directions are respectively associated with indices i, j and k. Thesubscripts on the � operator stands for the derivative coordinates of the di�eren-tial operator it approximates. The superscripts are the grid indices on which it iscentered. The operand appearing after the � operator is a discrete scalar variabledependent on the appropriate grid indices. The generic variable u is used here as anexample. Some examples of these �nite di�erence formulae areux � �xn;i;j;ku = un;i+1=2;j;k � un;i�1=2;j;kh (D.1)uxx � �xxn;i;j;ku = �xn;i+1=2;j;ku ��xn;i�1=2;j;kuh= un;i+1;j;k � 2un;i;j;k + un;i�1;j;kh2 (D.2)uxxx � �xxxn;i;j;ku = �xxn;i+1=2;j;ku ��xxn;i�1=2;j;kuh= un;i+3=2;j;k � 3un;i+1=2;j;k + 3un;i�1=2;j;k � un;i�3=2;j;kh3 (D.3)

235uxxxx � �xxxxn;i;j;ku = �xxxn;i+1=2;j;ku ��xxxn;i�1=2;j;kuh= un;i+2;j;k � 4un;i+1;j;k + 6un;i;j;k � 4un;i�1;j;k + un;i�2;j;kh4uxxxxx � �xxxxxn;i;j;ku = �xxxxn;i+1=2;j;ku ��xxxxn;i�1=2;j;kuh = � � � (D.4)uxyy � �xyyn;i;j;ku = �yyn;i+1=2;j;ku ��yyn;i�1=2;j;kuh= 1h3 ((un;i+1=2;j�1;k � 2un;i+1=2;j;k + un;i+1=2;j+1;k)= �(un;i�1=2;j�1;k � 2un;i�1=2;j;k + un;i�1=2;j+1;k)) (D.5)uxxyy � �xxyyn;i;j;ku = �xyyn;i+1=2;j;ku ��xyyn;i�1=2;j;kuh = � � � (D.6)uxyyzz � �xyyzzn;i;j;ku = �yyzzn;i+1=2;j;ku ��yyzzn;i�1=2;j;kuh = � � � (D.7)uxyyyy � �xyyyyn;i;j;ku = �yyyyn;i+1=2;j;ku ��yyyyn;i�1=2;j;kuh = � � � (D.8)These approximations are quite simple and their use is widespread.It should be noted that these formulae are accurate to O(h2), but due to thearrangement of the dependent variables on the grid, they do not always work, becausethey can require data that is not present. As it turns out, this does not pose anyproblem for standard leapfrog schemes. (These approximation formulae use onlydata that belong to the same decoupled solution, when the standard leapfrog schemeis viewed as one of the decoupled sets from a leapfrog solution on a doubly �ne grid.)But the high-order upwind leapfrog schemes of Chapter V do require derivatives forwhich the arrangement of dependent variables present the problem of missing data.For example, one cannot use the approximation Gxxxy = �xn;i;j;kGx when (i; j; k)corresponds to a cell center in the upwind leapfrog scheme. For these cases, wede�ne the alternative discrete di�erence operators e�. Let Gx be a variable de�nedonly on the x-faces and the index (i; j; k) coordinates be the center of such a face in

236the upwind leapfrog scheme. ThenGxxx � e�xxn;i;j;kGx= Gxn;i+3=2;j;k �Gxn;i+1=2;j;k �Gxn;i�1=2;j;k �Gxn;i�3=2;j;k2h2 (D.9)which di�ers from �xxn;i;j;kGx only by O(h2) but can be used when �xxn;i;j;kGx cannotbe. Other di�erence operators such as e�yyn;i;j;kGy and e�zzn;i;j;kGz follow suit.Another discretization that cannot use the standard � operator is, for example,Gxyz , which must use the formulaGxyz � e�yzn;i;j;kGx= Gxn;i+1;j+1;k �Gxn;i+1;j�1;k �Gyn;i�1;j+1;k +Gyn;i�1;j�1;k4h2 (D.10)which di�ers from �yzn;i;j;kGx only by O(h2) and can be used when �yzn;i;j;kGx cannotbe. Needless to say, these formulae are used in this dissertation only when they work.

BIBLIOGRAPHY

237

238BIBLIOGRAPHY[1] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961.[2] R. F. Harrington, Field Computations by Moment Methods. New York: Macmil-lan, 1968.[3] B. T. Nguyen and S. Hutchinson, \The implementation of the upwind leapfrogscheme for 3d electromagnetic scattering on massively parallel computers," Tech.Rep. SAND95-1322, Sandia, July 1995.[4] J. N. Shadid and R. S. Tuminaro, \Sparse iterative algorithm software for large-scale MIMD machines: An initial discussion and implementation," Tech. Rep.SAND91-0059, Sandia, 1991.[5] A. Ta ove and K. R. Umashankar, \Review of FD-TD numerical modeling ofelectromagnetic wave scattering and radar cross section," Proc. IEEE, vol. 77,pp. 682{699, May 1989.[6] V. Shankar, \Research to application{supercomputing trends for the 90's op-portunities for interdisciplinary computations," AIAA Paper 91-0002, 1991.[7] V. Shankar, A. Mohammadian, W. Hall, and R. Erickson, \CFD spino�{computational electromagnetics for radar cross section (RCS) studies," AIAAPaper 90-3055, 1990.[8] A. Ta ove, \Re-inventing electromagnetics: supercomputing solution ofMaxwell's equations via direct time integration on space grids," AIAA Paper92-0333, January 1992.[9] W. Hall. Personal Communication, 1995.[10] S. R. Friberg, A. M. Weiner, Y. Silberberg, B. G. Sfez, and P. S. Smith,\Femtosecond switching in a dual-core-�ber nonlinear coupler," Optics Letters,vol. 13, pp. 904{906, October 1988.[11] K. Blow and D. Wood, \Theoretical description of transient stimulated ramanscattering in optical �bers," IEEE Journal of Quantum Electronics, vol. 25,pp. 2665{2673, December 1989.

239[12] P. Pelet and N. Engheta, \The theory of chiral waveguides," IEEE Transactionson Antennas and Propagation, vol. 38, pp. 90{97, January 1990.[13] W. B. Scott, \Stealth/counter-stealth technologies," Aviation Week & SpaceTechnology, vol. 131, pp. 38{41, December 4 1989.[14] K. S. Yee, \Numerical solution of initial boundary value problems involvingMaxwell's equations in isotropic media," IEEE Transactions on Antennas andPropagation, vol. AP-14, pp. 302{307, May 1966.[15] A. Bayliss and E. Turkell, \Radiation boundary conditions for wave-like equa-tions," Communications on Pure and Applied Mathematics, vol. 33, pp. 707{725,1980.[16] R. L. Higdon, \Numerical absorbing boundary conditions for the wave equa-tion," Mathematics of Computation, vol. 49, pp. 65{90, July 1987.[17] B. Engquist and A. Majda, \Absorbing boundary conditions for the numericalsimulation of waves," Mathematics of Computation, vol. 31, 1977.[18] P. Monk and E. Suli, \A convergence analysis of Yee's scheme on nonuniformgries," Journal of Computational Physics, vol. 31, pp. 393{412, April 1994.[19] R. Holland, \Finite di�erence solutions of Maxwell's equations in generalizednonorthogonal coorinates," IEEE Transactions on Nuclear Science, vol. NS-30,pp. 4689{4591, December 1983.[20] N. K. Madsen and R. W. Ziolkowski, \Numerical solution of Maxwell's equationsin the time domain using irregular nonorthogonal grids," Wave Motion, vol. 10,pp. 583{596, December 1988.[21] M. Fusco, \FDTD algorithm in curvilinear coordinates," IEEE Transactions onAntennas and Propagation, vol. 38, pp. 76{89, January 1990.[22] M. A. Fusco, M. V. Smith, and L. W. Gordon, \A three-dimensional FDTD al-gorithm in curvilinear coordinates," IEEE Transactions on Antennas and Prop-agation, vol. 39, pp. 1463{1471, October 1991.[23] K. S. Yee, J. S. Chen, and A. H. Chang, \Conformal �nite-di�erence time-domain (FDTD) with overlaping grids," IEEE Transactions on Antennas andPropagation, vol. 40, pp. 1068{1075, September 1992.[24] J. A. Benek, J. L. Steger, and J. C. Dougherty, \A exible grid embeddingtechnique with application to the euler equations," AIAA Paper 83-1944, 1983.[25] F. C. Dougherty, J. A. Benek, and J. L. Steger, \On applications of chimeragrid schemes to store separation," Technical Memorandum TM-88193, NASA,October 1985.

240[26] J. Fang, Time Domain Computation for Maxwell's Equations. PhD thesis, Uni-versity of California, Berkeley, California, November 1989.[27] T. Deveze, \High order F.D.T.D. algorithm to reduce numerical dispersionand staircasing," in 10th Annual Review of Progress in Applied ComputationalElectromagnetics, pp. 61{68, Applied Computational Electromagnetics Society,March 1994.[28] M. Krumpholz, C. Huber, and P. Russer, \A �eld theoretical comparison ofFDTD and TLM," IEEE Transactions on Microwave Theory and Techniques,vol. 43, pp. 1935{1950, September 1995.[29] M. Krumpholz and L. P. Katehi, \MRTD: New time-domain schemes based onmultiresolution analysis," IEEE Transactions on Microwave Theory and Tech-niques, vol. 44, pp. 555{571, April 1996.[30] A. C. Cangellaris, C.-C. Lin, and K. K. Mei, \Point-matched time domain �niteelement methods for electromagnetic radiation and scattering," IEEE Transac-tions on Antennas and Propagation, vol. AP-35, pp. 1160{1173, October 1987.[31] V. Shankar, W. Hall, and A. Mohammadian, \A three-dimensional Maxwell'sequation solver for computation of scattering from layered media," IEEE Trans-actions on Magnetics, vol. 25, pp. 3098{3103, July 1989.[32] V. Shankar, W. Hall, and A. Mohammadian, \A time-domain di�erential solverfor electromagnetic scattering problems," Proc. IEEE, vol. 77, pp. 709{721, May1989.[33] J. S. Shang, \A fractional-step method for solving 3D, time-domain Maxwellequations," Journal of Computational Physics, vol. 118, pp. 109{119, April 1995.[34] S. K. Godunov, \A �nite-di�erence method for the numerical computation anddiscontinuous solutions of the equations of uid dynamics," MatematicheskiiSbornik, vol. 47, pp. 271{306, 1959.[35] B. van Leer, \Towards the ultimate conservation di�erence scheme I. the questof monotinicity," Lecture Notes in Physics, vol. 18, pp. 163{168, April 1973.[36] B. van Leer, \Towards the ultimate conservation di�erence scheme ii. monotonic-ity and conservation combined in a second-order scheme," Journal of Computa-tional Physics, vol. 14, pp. 361{370, April 1974.[37] I. G. Katayev, Electromagnetic Shock Waves. London: Ili�e Books, Ltd., 1966.[38] J. P. Boris and D. L. Book, \Flux-corrected transport. I. SHASTA, a uidtransport algorithm that works," Journal of Computational Physics, vol. 11,pp. 38{69, January 1973.

241[39] S. T. Zalesak, \Fully multidimensional ux-corrected transport algorithms for uids," Journal of Computational Physics, vol. 31, pp. 335{362, 1979.[40] S. R. Omick and S. P. Castillo, \A new �nite-di�erence time-domain algorithmfor accurate modeling of wide-band electromagnetic phenomena," IEEE Journalof Quantum Electronics, vol. 28, pp. 2416{2422, October 1992.[41] S. R. Omick, High Accuracy Transient Finite Di�erence and Finite ElementAlgorithms in Computational Electromagnetics. PhD thesis, New Mexico StateUniversity, Las Cruces, New Mexico, December 1993.[42] A. H. Mohammadian, V. Shankar, and W. Hall, \Computation of electromag-netic scattering and radiation using a time-domain �nite-volume discretizationprocedure," Computer Physics Communications, vol. 68, pp. 175{196, 1991.[43] P. Goorjian and A. Ta ove, \Computational modeling of nonlinear electromag-netic phenomena," AIAA Paper 92-0457, January 1992.[44] P. M. Goorjian, \Computational modeling of femtosecond optical solitons fromMaxwell's equations," IEEE Journal of Quantum Electronics, vol. 28, pp. 2416{2422, October 1992.[45] C. K. W. Tam and J. C. Webb, \Dispersion-relation-preserving �nite di�er-ence schemes for computational acoustics," Journal of Computational Physics,vol. 107, pp. 262{281, 1993.[46] K. L. Shlager, J. G. Maloney, S. L. Ray, and A. F. Peterson, \Relative accuracyof several �nite-di�erence time-domain methods in two and three dimensions,"IEEE Transactions on Antennas and Propapation, vol. 41, pp. 1732{1737, De-cember 1993.[47] J. P. Thomas, C. Kim, and P. L. Roe, \Progress towards a new computationalscheme for aeroacoustics," AIAA Paper 95-1758, 1995.[48] P. L. Roe, \Linear bicharacteristic schemes without dissipation," Tech. Rep.94-65, ICASE, July 1994.[49] A. Iserles, \Generalised leapfrog schemes," IMA Journal of Numerical Analysis,vol. 6, 1986.[50] J. P. Thomas, An Investigation of the Upwind Leapfrog Method for Scalar Ad-vection and Acoustic/Aeroacoustic Wave Propagation Problems. PhD thesis,University of Michigan, Ann Arbor, MI, 1996.[51] J. von Neumann and R. D. Richtmyer, \A method for the numerical calculationof hydrodynamic shocks," Journal of Applied Physics, vol. 21, pp. 232{237,March 1950.[52] A. Iserles, Order Stars. London and New York: Chapman and Hall, 1991.

242[53] R. Courant, K. Friedrichs, and J. von Neumann, \Uber die partiellen di�eren-zgleichungen der matthematischen physik," Mathematische Annalen, vol. 100,pp. 32{74, 1928. English translation in IBM Journal (1967), 215-34.[54] J. P. Thomas and P. L. Roe, \Development of non-dissipative numerical schemesfor computational aeroacoustics," AIAA Paper 93-3382, 1993.[55] B. T. Nguyen and P. L. Roe, \Electromagnetic wave scattering computations us-ing an upwind leapfrog scheme," in Second International Conference and Work-shop on approximations & Numerical Methods for the Solution of the MaxwellEquations, (Washington, DC), George Washington University, October 1993.[56] P. D. Lax and B. Wendro�, \Systems of conservation laws," Communicationsin Pure and Applied Mathematics, vol. 13, pp. 217{237, 1960.[57] C. Hirsch, Numerical Computational of Internal and External Flows, vol. 1:Fundamentals of Numerical Discretization. New York: John Wiley & Sons,1988.

ABSTRACTINVESTIGATION OF THREE-LEVEL FINITE-DIFFERENCE TIME-DOMAINMETHODS FOR MULTIDIMENSIONAL ACOUSTICS ANDELECTROMAGNETICSbyBrian Thao NguyenChair: P.L. RoeThis dissertation deals with accurate �nite-di�erence time-domain methods fore�ciently solving acoustics and electromagnetic scattering problems. Problems in1D, 2D and 3D are treated. Primary emphasis is given to comparing variations ofthe existing standard leapfrog schemes and new upwind leapfrog schemes and tothe e�ectiveness of their high-order extensions. Numerical errors are described forthe schemes at given resolutions. Afterward, cost-e�ciency is compared betweenschemes dealing with the same problem. In 1D, additional comparison with di�usivecentral-di�erence and upstream-biased schemes are made to gain a perspective onthe advantages and disadvantages of non-di�usive schemes.The multi-dimensional upwind leapfrog schemes are derived on the basis of thoseoriginally developed for the 1D advection equation, as many upstream-biased schemes

are. To extend the 1D schemes for the multi-dimensional acoustics and electromag-netics equations, the equations are �rst written in bicharacteristic forms. Relat-ing the bicharacteristics equations to the 1D advection equation shows that multi-dimensional upwind leapfrog schemes can be derived by including terms that couplethe equations in the di�erent directions and by using an appropriate layout of vari-ables on the grid.All schemes surpass the widely-used second-order accurate standard leapfrogscheme in most respect, but the fourth-order extension of this scheme is very com-petitive. A very tight error tolerance is required to make the sixth-order accurateversion feasible. Higher-order schemes are always more memory-e�cient than lower-order versions, but their large computational stencils are less desirable than the morecompact lower-order stencils. The upwind leapfrog schemes always uses a smallerstencil than the standard leapfrog scheme stencils, for a given order of accuracy.The 2D fourth-order accurate upwind leapfrog scheme is a highly desirable schemein most respects. In 3D, the schemes are more closely matched, with the high-orderstandard leapfrog schemes being better in terms of overall e�ciency, though theupwind leapfrog schemes are not grossly ine�cient in comparison. No stable high-order upwind leapfrog scheme for electromagnetics has been found.Costs range over nearly two orders of magnitude between the least and moste�cient schemes, considering only the bare schemes. However, depending on theparticulars of the actual problem being solved, such as the presence of inhomogeneity,the results of the e�ciency comparison may change.