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# leveque_TOC - i i i rjlfdm page iii i Contents Preface I 1...

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“rjlfdm” 2007/4/12 page iii i i i i i i i Contents Preface ix I Boundary Value Problems and Iterative Methods 1 1 Finite difference approximations 3 1.1 Truncation errors . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Deriving finite difference approximations . . . . . . . . . . . . . 7 1.3 Second order derivatives . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Higher order derivatives . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 A general approach to deriving the coefficients . . . . . . . . . . 10 2 Steady States and Boundary Value Problems 13 2.1 The heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 The steady-state problem . . . . . . . . . . . . . . . . . . . . . . 14 2.4 A simple finite difference method . . . . . . . . . . . . . . . . . 15 2.5 Local truncation error . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6 Global error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.7 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.8 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.9 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.10 Stability in the 2-norm . . . . . . . . . . . . . . . . . . . . . . . 20 2.11 Green’s functions and max-norm stability . . . . . . . . . . . . . 22 2.12 Neumann boundary conditions . . . . . . . . . . . . . . . . . . . 30 2.13 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . 33 2.14 Ordering the unknowns and equations . . . . . . . . . . . . . . . 34 2.15 A general linear second order equation . . . . . . . . . . . . . . 35 2.16 Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.16.1 Discretization of the nonlinear BVP . . . . . . . . . 39 2.16.2 Nonuniqueness . . . . . . . . . . . . . . . . . . . . . 41 2.16.3 Accuracy on nonlinear equations . . . . . . . . . . . 42 2.17 Singular perturbations and boundary layers . . . . . . . . . . . 44 2.17.1 Interior layers . . . . . . . . . . . . . . . . . . . . . . 47 iii

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“rjlfdm” 2007/4/12 page iv i i i i i i i iv Contents 2.18 Nonuniform grids . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.18.1 Adaptive mesh selection . . . . . . . . . . . . . . . . 53 2.19 Continuation methods . . . . . . . . . . . . . . . . . . . . . . . 53 2.20 Higher order methods . . . . . . . . . . . . . . . . . . . . . . . . 54 2.20.1 Fourth order differencing . . . . . . . . . . . . . . . 54 2.20.2 Extrapolation methods . . . . . . . . . . . . . . . . 54 2.20.3 Deferred corrections . . . . . . . . . . . . . . . . . . 56 2.21 Spectral methods . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3 Elliptic Equations 61 3.1 Steady-state heat conduction . . . . . . . . . . . . . . . . . . . . 61 3.2 The five-point stencil for the Laplacian . . . . . . . . . . . . . . 62 3.3 Ordering the unknowns and equations . . . . . . . . . . . . . . . 63 3.4 Accuracy and stability . . . . . . . . . . . . . . . . . . . . . . . 65 3.5 The nine-point Laplacian . . . . . . . . . . . . . . . . . . . . . . 66 3.6 More difficult problems . . . . . . . . . . . . . . . . . . . . . . . 68 3.7 Solving the linear system . . . . . . . . . . . . . . . . . . . . . . 68 3.7.1 Sparse storage in matlab . . . . . . . . . . . . . . . 70 4 Iterative Methods for Sparse Linear Systems 71 4.1 Jacobi and Gauss-Seidel . . . . . . . . . . . . . . . . . . . . . . 71 4.2 Analysis of matrix splitting methods . . . . . . . . . . . . . . . 74 4.2.1 Rate of convergence . . . . . . . . . . . . . . . . . . 76 4.2.2 SOR . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3 Descent methods and conjugate gradients . . . . . . . . . . . . . 80 4.3.1 The method of steepest descent . . . . . . . . . . . . 81 4.3.2 The A -conjugate search direction . . . . . . . . . . . 85 4.3.3 The conjugate-gradient algorithm . . . . . . . . . . 89 4.3.4 Convergence of CG . . . . . . . . . . . . . . . . . . 91 4.3.5 Preconditioners . . . . . . . . . . . . . . . . . . . . . 97 4.3.6
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