LECTURES on COMPUTATIONAL NUMERICAL ANALYSIS of PARTIAL DIFFERENTIAL EQUATIONS J. M. McDonough Departments of Mechanical Engineering and Mathematics University of Kentucky c 1985, 2002, 2008 Contents 1 Introduction 1 1.1 Basic Mathematical Definitions . 2 1.1.1 Norms and related ideas . 2 1.1.2 Convergence of sequences . 5 1.1.3 Consistency, stability and convergence . 7 1.2 Classifications of Partial Differential Equations . 10 1.2.1 Equation type . 10 1.2.2 Form of nonlinearity . 11 1.3 Well Posedness of PDE Problems . 12 1.4 Discretization and Gridding of PDE Problems . 13 1.4.1 Discretization techniques . 13 1.4.2 Gridding methods . 15 1.5 Summary . 16 2 Numerical Solution of Elliptic Equations 17 2.1 Background . 19 2.1.1 Iterative solution of linear systems|an overview . 19 2.1.2 Basic theory of linear iterative methods . 20 2.2 Successive Overrelaxation . 25 2.2.1 Jacobi iteration . 25 2.2.2 SOR theory . 28 2.2.3 Some modifications to basic SOR . 34 2.3 Alternating Direction Implicit (ADI) Procedures . 40 2.3.1 ADI with a single iteration parameter . 41 2.3.2 ADI: the commutative case . 45 2.3.3 ADI: the noncommutative case . 49 2.4 Incomplete LU Decomposition (ILU) . 49 2.4.1 Basic ideas of ILU decomposition . 49 2.4.2 The strongly implicit procedure (SIP) . 51 2.5 Preconditioning . 57 2.6 Conjugate-Gradient Acceleration . 59 2.6.1 The method of steepest descent . 59 2.6.2 Derivation of the conjugate-gradient method . 61 2.6.3 Relationship of CG to other methods . 63 2.7 Introduction to Multigrid Procedures . 64 2.7.1 Some basic ideas . 64 2.7.2 The h-2h two-grid algorithm . 66 2.7.3 `-grid multigrid methods . 70 2.7.4 The full multigrid method . 71 2.7.5 Some concluding remarks . 74 i CONTENTS i 2.8 Domain-Decomposition Methods . 75 2.8.1 The alternating Schwarz procedure . 77 2.8.2 The Schur complement . 79 2.8.3 Multiplicative and additive Schwarz methods . 81 2.8.4 Multilevel domain-decomposition methods . 87 2.9 Summary . 89 3 Time-Splitting Methods for Evolution Equations 91 3.1 Alternating Direction Implicit Methods . 92 3.1.1 Peaceman{Rachford ADI . 92 3.1.2 Douglas{Rachford ADI . 95 3.1.3 Implementation of ADI schemes . 96 3.2 Locally One-Dimensional Methods . 98 3.3 General Douglas{Gunn Procedures . 102 3.3.1 D{G methods for two-level difference equations . 102 3.3.2 D{G methods for multi-level difference equations . 108 3.4 Summary . 111 4 Various Miscellaneous Topics 113 4.1 Nonlinear PDEs . 113 4.1.1 The general nonlinear problem to be considered . 113 4.1.2 Explicit integration of nonlinear terms . 113 4.1.3 Picard iteration . 114 4.1.4 The Newton{Kantorovich Procedure . 114 4.2 Systems of PDEs . 119 4.2.1 Example problem|a generalized transport equation . 119 4.2.2 Quasilinearization of systems of PDEs . 120 4.3 Numerical Solution of Block-Banded Algebraic Systems . 123 4.3.1 Block-banded LU decomposition|how it is applied . 123 4.3.2 Block-banded LU decomposition details . 124 4.3.3 Arithmetic operation counts . 126 4.4 Cell-Re and Aliasing Treatments . 127 4.4.1 The cell-Re problem|its definition and treatment . 127 4.4.2 Treatment of effects of aliasing . 136 4.5 More Advanced Spatial Discretizations . 142 4.5.1 Basic approximations for mixed derivatives . 143 4.5.2 Mixed derivatives with variable coefficients . 144 4.5.3 Discretization of self-adjoint form second (unmixed) derivatives . 147 4.6 Treatment of Advanced Boundary Conditions . 150 4.6.1 General linear BCs . 151 4.6.2 Outflow BCs . 152 4.7 Summary . 153 5 Numerical Solution of Partial Differential Equations on Irregular Domains|Grid Gen- eration 155 5.1 Overview of Grid Structures . 155 5.1.1 Unstructured grids . 156 5.1.2 Structured grids . 158 References 159 ii CONTENTS List of Figures 1.1 Schematic of a contraction mapping, [a; b] [f(a); f(b)]. 7 ! 1.2 Methods for spatial discretization of partial differential equations; (a) finite difference, (b) finite element and (c) spectral. 14 2.1 N N {point grid and mesh star for discretizations of Eq. (2.1). 18 x × y 2.2 Sparse, banded matrices arising from finite-difference discretizations of elliptic operators: (a) 5-point discrete Laplacian; (b) 9-point general discrete elliptic operator. 18 2.3 Qualitative comparison of required arithmetic for various iterative methods for 2-D elliptic problems. 20 2.4 Qualitative representation of error reduction during linear fixed-point iterations. 23 2.5 Discretization of the Laplace/Poisson equation on a rectangular grid of N N points. 26 x × y 2.6 Band structure of Jacobi iteration matrix for Laplace/Poisson equation. 27 2.7 Geometric test of consistent ordering. (a) consistent ordering, (b) nonconsistent ordering. 31 2.8 Spectral radius of SOR iteration matrix vs. !. 32 2.9 Red-black ordering for discrete Laplacian. 35 2.10 Comparison of computations for point and line SOR showing grid stencils and red-black ordered lines. 37 2.11 Matrices arising from decomposition of A: (a) H matrix, (b) V matrix, (c) S matrix. 41 2.12 (a) 7-band finite-difference matrix; (b) corresponding mesh star. 51 2.13 Finite-difference grid for demonstrating structure of SIP matrices. 52 2.14 Level set contours and steepest-descent trajectory of 2-D quadratic form. 60 2.15 Level set contours, steepest-descent trajectory and conjugate gradient trajectory of 2-D quadratic form. 61 2.16 Comparison of h and 2h grids for multigrid implementations. 68 2.17 Multigrid V-cycles; (a) ` = 2, and (b) ` = 3. 71 2.18 Multigrid V-cycles with ` = 3 and different values of γ; (a) γ = 1, (b) γ = 2 and (c) γ = 3. 71.