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LECTURES onCOMPUTATIONAL NUMERICALANALYSISofPARTIAL DIFFERENTIAL EQUATIONSJ. M. McDonoughDepartments of Mechanical Engineering and MathematicsUniversity of Kentuckyc (1985, 2002iiContents1 Numerical Solution of Elliptic Equations 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Iterative solution of linear systemsan overview . . . . . . . . . . . . . . . . 21.1.2 Basic theory of linear iterative methods . . . . . . . . . . . . . . . . . . . . . 51.2 Successive Overrelaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.1 Jacobi iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.2 SOR theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.3 Some modications to basic SOR . . . . . . . . . . . . . . . . . . . . . . . . . 191.3 Alternating Direction Implicit (ADI) Procedures . . . . . . . . . . . . . . . . . . . . 251.3.1 ADI with a single iteration parameter . . . . . . . . . . . . . . . . . . . . . . 261.3.2 ADI: the commutative case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.3.3 ADI: the noncommutative case . . . . . . . . . . . . . . . . . . . . . . . . . . 351.4 Incomplete LU Decomposition (ILU) . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.4.1 Basic ideas of ILU decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 361.4.2 The strongly implicit procedure (SIP) . . . . . . . . . . . . . . . . . . . . . . 371.5 Preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.6 Conjugate Gradient Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461.6.1 The method of steepest descent . . . . . . . . . . . . . . . . . . . . . . . . . . 461.6.2 Derivation of the conjugate gradient method . . . . . . . . . . . . . . . . . . 481.6.3 Relationship of CG to other methods . . . . . . . . . . . . . . . . . . . . . . . 501.7 Introduction to Multigrid Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . 511.7.1 Some basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511.7.2 The h-2h two-grid algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 531.7.3 -grid multigrid methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581.7.4 The full multigrid method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591.7.5 Some concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621.8 Domain Decomposition Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631.8.1 The alternating Schwarz procedure . . . . . . . . . . . . . . . . . . . . . . . . 651.8.2 The Schur complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681.8.3 Multiplicative and additive Schwarz methods . . . . . . . . . . . . . . . . . . 701.8.4 Multilevel domain decomposition methods . . . . . . . . . . . . . . . . . . . . 752 Time-Splitting Methods for Evolution Equations 792.1 Alternating Direction Implicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . 802.1.1 Peaceman-Rachford ADI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.1.2 Douglas-Rachford ADI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832.1.3 Implementation of ADI schemes . . . . . . . . . . . . . . . . . . . . . . . . . 84iiiCONTENTS i2.2 Locally One-Dimensional Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.3 General Douglas-Gunn Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902.3.1 D-G methods for two-level dierence equations . . . . . . . . . . . . . . . . . 902.3.2 D-G methods for multi-level dierence equations . . . . . . . . . . . . . . . . 963 Various Miscellaneous Topics 1013.1 Nonlinear PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.1.1 The general nonlinear problem to be considered . . . . . . . . . . . . . . . . . 1013.1.2 Explicit integration of nonlinear terms . . . . . . . . . . . . . . . . . . . . . . 1013.1.3 Picard iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.1.4 The Newton-Kantorovich Procedure . . . . . . . . . . . . . . . . . . . . . . . 1023.2 Systems of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.2.1 Example problema generalized transport equation . . . . . . . . . . . . . . 1073.2.2 Quasilinearization of systems of PDEs . . . . . . . . . . . . . . . . . . . . . . 1083.3 Numerical Solution of Block-Banded Algebraic Systems . . . . . . . . . . . . . . . . 1113.3.1 Block-banded LU decompositionhow it is applied . . . . . . . . . . . . . . . 1113.3.2 Block-banded LU decomposition details . . . . . . . . . . . . . . . . . . . . . 1123.3.3 Arithmetic operation counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114References 116ii CONTENTSList of Figures1.1 NxNypoint grid and mesh star for discretizations of Eq. (1.1). . . . . . . . . . . . 21.2 Sparse, banded matrices arising from nite-dierence discretizations of elliptic oper-ators: (a) 5-point discrete Laplacian; (b) 9-point general discrete elliptic operator. . 31.3 Qualitative comparison of required arithmetic for various iterative methods. . . . . . 41.4 Qualitative representation of error reduction during linear xed-point iterations. . . 71.5 Discretization of the Laplace/Poisson equation on a rectangular grid of NxNy points. 101.6 Band structure of Jacobi iteration matrix for Laplace/Poisson equation. . . . . . . . 111.7 Geometric test of consistent ordering. (a) consistent ordering, (b) nonconsistentordering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.8 Spectral radius of SOR iteration matrix vs. . . . . . . . . . . . . . . . . . . . . . . . 181.9 Red-black ordering for discrete Laplacian. . . . . . . . . . . . . . . . . . . . . . . . . 201.10 Comparison of computations for point and line SOR showing grid stencils and red-black ordered lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.11 Matrices arising from decomposition of A: (a) H matrix, (b) V matrix, (c) S matrix. 271.12 (a) 7-band nite-dierence matrix; (b) corresponding mesh star. . . . . . . . . . . . 381.13 Finite-dierence grid for demonstrating structure of SIP matrices. . . . . . . . . . . 381.14 Level set contours and steepest descent trajectory of 2-D quadratic form. .

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