About the Burton-Miller factor in the low frequency region

TL;DR

This paper investigates the Burton-Miller method's performance at low frequencies, compares different coupling parameters through numerical experiments, and proposes an enhancement technique using modified Richardson iteration for better stability and accuracy.

Contribution

The study introduces alternative coupling parameter choices for the Burton-Miller method in low frequency regions and proposes an iterative enhancement to improve solution stability.

Findings

Different parameter choices reduce condition number and error at low frequencies.

Numerical experiments demonstrate the effectiveness of alternative parameters.

Modified Richardson iteration improves solution stability in low frequency regime.

Abstract

The Burton-Miller method is a widely used approach in acoustics to enhance the stability of the boundary element method for exterior Helmholtz problems at so-called critical frequencies. This method depends on a coupling parameter $\eta$ and it can be shown that as long as $\eta$ has an imaginary part different from 0, the boundary integral formulation for the Helmholtz equation has a unique solution at all frequencies. A popular choice for this parameter is $\eta = \frac{\mathrm{i}}{k}$, where $k$ is the wavenumber. It can be shown that this choice is quasi optimal, at least in the high frequency limit. However, especially in the low frequency region, where the critical frequencies are still sparsely distributed, different choices for this factor result in a smaller condition number and a smaller error of the solution. In this work, alternative choices for this factor are compared based on numerical experiments. Additionally, a way to enhance the Burton-Miller solution with $\eta = \frac{\mathrm{i}}{k}$ for a sound hard scatterer in the low frequency region by an additional step of a modified Richardson iteration is introduced.