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leveque_TOC - rjlfdm 2007/4/12 page iii i i i i i i i i...

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Unformatted text preview: rjlfdm 2007/4/12 page iii i 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 Greens 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 rjlfdm 2007/4/12 page iv i 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 . . . . . . . . . . . . . . .Ordering the unknowns and equations ....
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This note was uploaded on 08/30/2011 for the course APMA 4301 taught by Professor Keyes during the Fall '08 term at Columbia.

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leveque_TOC - rjlfdm 2007/4/12 page iii i i i i i i i i...

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