# Convergence of finite difference methods for the wave equation in two space dimensions

## TL;DR

This paper extends normal mode analysis to two-dimensional wave equations solved by finite difference methods, providing a framework to understand boundary truncation errors and their impact on convergence, supported by numerical experiments.

## Contribution

It introduces a general framework for analyzing convergence of finite difference methods for 2D wave equations, including boundary and corner truncation errors, extending previous 1D analyses.

## Key findings

- Boundary truncation errors affect overall convergence.
- The analysis applies to second order wave equations.
- Numerical results confirm theoretical predictions.

## Abstract

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.