Numerical solution of PDEs

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<p>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. . . . . . . 471.15 Level set contours, steepest descent trajectory and conjugate gradient trajectory of2-D quadratic form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481.16 Comparison of h and 2h grids for multigrid implementations. . . . . . . . . . . . . . 551.17 Multigrid V-cycles; (a) = 2, and (b) = 3. . . . . . . . . . . . . . . . . . . . . . . . 581.18 Multigrid V-cycles with = 3 and dierent values of ; (a) = 1, (b) = 2 and (c) = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591.19 Four-Level, V-cycle full multigrid schematic. . . . . . . . . . . . . . . . . . . . . . . . 601.20 L-shaped grid depicting basic domain decomposition approach. . . . . . . . . . . . . 641.21 Keyhole-shaped domain 1 2 considered by Schwarz [35]. . . . . . . . . . . . . . . 651.22 Simple two subdomain problem to demonstrate Schur complement. . . . . . . . . . . 681.23 Simple two subdomain problem to demonstrate Schur complement. . . . . . . . . . . 691.24 Domain decomposition with two overlapping subdomains; (a) domain geometry, (b)matrix structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701.25 Schematic depicting two-level domain decomposition and approximate Schur com-plement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762.1 Implementation of line-by-line solves for time splitting of time-dependent problems. . 852.2 Numerical Dirichlet-Neumann problem; points on dashed lines are image pointsneeded for implementation of centered discretizations. . . . . . . . . . . . . . . . . . 861Chapter 1Numerical Solution of Elliptic EquationsIn this chapter we will study the solution of linear elliptic partial dierential equations (PDEs) vianumerical techniques. These equations typically represent steady-state physical situations, and intwo space dimensions (2D) assume the general form(a1(x, y)ux)x + (a2(x, y)uy)y +(a3(x, y)ux)y + (a4(x, y)uy)x+(a5(x, y)u)x + (a6(x, y)u)y +a7u(x, y) = f(x, y) (1.1)on a domain R2with appropriate boundary conditions, e.g., combinations of Dirichlet,Neumann and Robin) prescribed on . Here, subscripts denote partial dierentiation; e.g.,ux = u/x. It will be assumed that the coecients of (1.1) are such as to render the PDEelliptic, uniformly in .Throughout these lectures we will employ straightforward second-order centered nite-dierenceapproximations of derivatives (with an occasional exception), primarily for simplicity and ease ofpresentation. Applying such discretization to Eq. (1.1) results in a system of algebraic equations,A(1)i,j ui1,j1 +A(2)i,j ui1,j +A(3)i,j ui1,j+1 +A(4)i,j ui,j1 +A(5)i,j ui,j+A(6)i,j ui,j+1 +A(7)i,j ui+1,j1 +A(8)i,j ui+1,j +A(9)i,j ui+1,j+1 = fi,j , (1.2)i = 1, . . . , Nx, j = 1, . . . , Ny .We note that boundary conditions are assumed to have been included in this system of equations,so this corresponds to a solution on a NxNypoint grid, including boundary points, as depictedin Fig. 1.1. We have also indicated in this gure the mesh star corresponding to Eq. (1.2).We should comment here that while we will essentially always be concerned with 2-D problemsin these lectures, this is done merely for simplicity of presentation. Nearly all of the numerical al-gorithms to be considered apply equally well in 3D, and this can be assumed unless we specicallynote otherwise. Moreover, although as already noted, we will employ nite-dierence discretiza-tions, most of the solution methods to be discussed also apply for nite-element methods (FEMs).The system of linear algebraic equations corresponding to Eqs. (1.2) is sparse and banded, asshown in Fig. 1.2. Part (a) of this gure corresponds to a second-order centered discretization ofa Laplace operator, uxx + uyy, while part (b) is associated with the more general operator of Eq.(1.1). We will typically employ the concise notationAu = b (1.3)to represent...</p>

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