Finite Difference Schemes and Partial Differential EquationsThis book combines practical aspects of implementation with theoretical analysis of finite difference schemes and partial differences schemes. There is a thorough discussion of the concepts of convergence, consistency, and stability for time-dependent equations. The von Neumann analysis of stability is developed rigorously using the methods of Fourier analysis. Fourier analysis is used throughout the text, providing a unified treatment of the basic concepts and results. A complete proof of the Lax-Richtmyer theorem for equations with constant coefficients is included. |
Contents
Hyperbolic Partial Differential Equations | 1 |
Analysis of Finite Difference Schemes | 32 |
Order of Accuracy of Finite Difference Schemes | 53 |
Copyright | |
16 other sections not shown
Common terms and phrases
a²X² accurate of order accurate scheme algorithm amplification factor analysis approximation boundary conditions bounded coefficients computed conjugate gradient method consider convergence cos² Crank-Nicolson scheme defined derivative discussion eigenvalues elliptic equations equal equivalent error estimate example Exercise finite difference scheme five-point Laplacian formula Fourier transform Gauss-Seidel method given grid points grid spacing heat equation hyperbolic equations initial data initial function initial value problem initial-boundary value problem integral Jacobi method Kreiss L2 norm Laplacian Lax-Friedrichs scheme Lax-Wendroff scheme leapfrog scheme lemma linear multistep schemes Neumann polynomial obtain one-step scheme one-way wave equation order of accuracy Parseval's relation partial differential equations pk+1 positive definite preconditioned proof prove roots satisfies scheme is stable Schur polynomial second-order accurate Section Show sin² solve stability condition Taylor series tend to zero Theorem u(tn un+1 variable ve,m vector vn+1 well-posed αλ