Finite Difference Methods for Ordinary and Partial Differential Equations- Steady-State and Time-Dependent Problems (Classics in Applied Mathematics).pdf

文档介绍：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 plete and may contain errors.
They are made available primarily for students in my courses.
Please contact me for other uses.
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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
Truncation errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Deriving finite difference approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Polynomial interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Second order derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Higher order derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Boundary Value Problems 11
The heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
The steady-state problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
A simple finite difference method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Local truncation error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Global error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Consistency . . . . . . . . . . . . . . . . . . . . . . . .