# Boundary integral equation methods for the two dimensional wave equation in time domain revisited

## TL;DR

This paper analyzes the stability of time domain boundary integral equation methods for the 2D wave equation, proposing a numerical approach to identify and reformulate unstable equations for improved stability.

## Contribution

It introduces a novel stability analysis linking time domain BIEMs to nonlinear eigenvalue problems and offers reformulations for stability enhancement.

## Key findings

- Stability reduces to a nonlinear eigenvalue problem.
- Some frequency domain resonance-free equations cause instability in time domain.
- Reformulating equations can lead to stable numerical schemes.

## Abstract

This study considers the stability of time domain BIEMs for the wave equation in 2D. We show that the stability of time domain BIEMs is reduced to a nonlinear eigenvalue problem related to frequency domain integral equations. We propose to solve this non-linear eigenvalue problem numerically with the Sakurai-Sugiura method. After validating this approach numerically in the exterior Dirichlet problem, we proceed to transmission problems in which we find that some time domain counterparts of "resonance-free" integral equations in frequency domain lead to instability. We finally show how to reformulate these equations to obtain stable numerical schemes.