Convergence of finite difference methods for the wave equation in two space dimensions
For researchers using finite difference methods for wave equations in 2D, this provides a framework to analyze boundary-induced convergence issues, though it is an incremental extension of existing 1D analysis.
This paper extends normal mode analysis for finite difference methods from one to two space dimensions, analyzing how lower-order truncation errors near boundaries affect convergence rates. The analysis covers errors along entire boundaries and near corners, with numerical experiments confirming the theoretical results.
When using a finite difference method to solve an initial--boundary--value problem, the truncation error is often of lower order at a few grid points near boundaries than in the interior. Normal mode analysis is a powerful tool to analyze the effect of the large truncation error near boundaries on the overall convergence rate, and has been used in many previous literatures for different equations. However, existing work only concerns problems in one space dimension. In this paper, we extend the analysis to problems in two space dimensions. The two dimensional analysis is based on a diagonalization procedure that decomposes a two dimensional problem to many one dimensional problems of the same type. We present a general framework of analyzing convergence for such one dimensional problems, and explain how to obtain the result for the corresponding two dimensional problem. In particular, we consider two kinds of truncation errors in two space dimensions: the truncation error along an entire boundary, and the truncation error localized at a few grid points close to a corner of the computational domain. The accuracy analysis is in a general framework, here applied to the second order wave equation. Numerical experiments corroborate our accuracy analysis.