Improved Error Bounds for Dirichlet-to-Neumann Absorbing Boundaries

TL;DR

This paper improves the theoretical error bounds for Dirichlet-to-Neumann absorbing boundary conditions in wave equations, demonstrating that the error grows quadratically with time rather than exponentially, thus providing more accurate and reliable boundary approximations.

Contribution

The authors establish new rigorous error bounds showing quadratic growth in time, enhancing understanding of the accuracy of Dirichlet-to-Neumann boundary conditions.

Findings

Error behaves like epsilon t^2 instead of exponential in time

Provides more precise error estimates for wave boundary conditions

Improves theoretical understanding of absorbing boundary accuracy

Abstract

It has long been known how to construct radiation boundary conditions for the time dependent wave equation. Although arguments suggesting that they are accurate have been given, it is only recently that rigorous error bounds have been proved. Previous estimates show that the error caused by these methods behaves like epsilon C exp(gamma t) for any gamma > 0. We improve these results and show that the error behaves like epsilon t^2.