Finite Difference Methods for Differential Equations Randall J. LeVeque DRAFT VERSION for use in the course AMath 585{586 University of Washington Version of September, 2005 WARNING: These notes are incomplete and may contain errors. They are made available primarily for students in my courses. Please contact me for other uses. [email protected] c R. J. LeVeque, 1998{2005 2 c R. J. LeVeque, 2004 | University of Washington | AMath 585{6 Notes Contents I Basic Text 1 1 Finite difference approximations 3 1.1 Truncation errors . 5 1.2 Deriving finite difference approximations . 6 1.3 Polynomial interpolation . 7 1.4 Second order derivatives . 7 1.5 Higher order derivatives . 8 1.6 Exercises . 8 2 Boundary Value Problems 11 2.1 The heat equation . 11 2.2 Boundary conditions . 12 2.3 The steady-state problem . 12 2.4 A simple finite difference method . 13 2.5 Local truncation error . 14 2.6 Global error . 15 2.7 Stability . 15 2.8 Consistency . 16 2.9 Convergence . 16 2.10 Stability in the 2-norm . 17 2.11 Green's functions and max-norm stability . 19 2.12 Neumann boundary conditions . 21 2.13 Existence and uniqueness . 23 2.14 A general linear second order equation . 24 2.15 Nonlinear Equations . 26 2.15.1 Discretization of the nonlinear BVP . 27 2.15.2 Nonconvergence . 29 2.15.3 Nonuniqueness . 29 2.15.4 Accuracy on nonlinear equations . 29 2.16 Singular perturbations and boundary layers . 31 2.16.1 Interior layers . 33 2.17 Nonuniform grids and adaptive refinement . 34 2.18 Higher order methods . 34 2.18.1 Fourth order differencing . 34 2.18.2 Extrapolation methods . 35 2.18.3 Deferred corrections . 36 2.19 Exercises . 37 i ii CONTENTS 3 Elliptic Equations 39 3.1 Steady-state heat conduction . 39 3.2 The five-point stencil for the Laplacian . 40 3.3 Accuracy and stability . 43 3.4 The nine-point Laplacian . 44 3.5 Solving the linear system . 45 3.5.1 Gaussian elimination . 45 3.5.2 Fast Poisson solvers . 46 3.6 Exercises . 48 4 Function Space Methods 49 4.1 Collocation . 49 4.2 Spectral methods . 50 4.2.1 Matrix interpretation . 52 4.2.2 Accuracy . 52 4.2.3 Stability . 53 4.2.4 Collocation property . 53 4.2.5 Pseudospectral methods based on polynomial interpolation . 53 4.3 The finite element method . 56 4.3.1 Two space dimensions . 59 4.4 Exercises . 60 5 Iterative Methods for Sparse Linear Systems 61 5.1 Jacobi and Gauss-Seidel . 61 5.2 Analysis of matrix splitting methods . 63 5.2.1 Rate of convergence . 65 5.2.2 SOR . 66 5.3 Descent methods and conjugate gradients . 67 5.3.1 The method of steepest descent . 69 5.3.2 The A-conjugate search direction . 74 5.3.3 The conjugate-gradient algorithm . 76 5.3.4 Convergence of CG . 78 5.3.5 Preconditioners . 84 5.4 Multigrid methods . 86 6 The Initial Value Problem for ODE's 93 6.1 Lipschitz continuity . 94 6.1.1 Existence and uniqueness of solutions . 95 6.1.2 Systems of equations . 96 6.1.3 Significance of the Lipschitz constant . 96 6.1.4 Limitations . 97 6.2 Some basic numerical methods . 98 6.3 Truncation errors . 99 6.4 One-step errors . 99 6.5 Taylor series methods . 100 6.6 Runge-Kutta Methods . 101 6.7 1-step vs. multistep methods . 103 6.8 Linear Multistep Methods . 104 6.8.1 Local truncation error . 105 6.8.2 Characteristic polynomials . 106 6.8.3 Starting values . 106 6.9 Exercises . 107 R. J. LeVeque | AMath 585{6 Notes iii 7 Zero-Stability and Convergence for Initial Value Problems 109 7.1 Convergence . 109 7.2 Linear equations and Duhamel's principle . 110 7.3 One-step methods . 110 7.3.1 Euler's method on linear problems . 110 7.3.2 Relation to stability for BVP's . 112 7.3.3 Euler's method on nonlinear problems . 113 7.3.4 General 1-step methods . 113 7.4 Zero-stability of linear multistep methods . 114 7.4.1 Solving linear difference equations . 115 7.5 Exercises . 118 8 Absolute Stability for ODEs 119.