Hybrid asymptotic/numerical methods for the evaluation of layer heat potentials in two dimensions

TL;DR

This paper introduces a hybrid asymptotic/numerical approach for accurately computing layer heat potentials in two dimensions, overcoming stiffness issues and enabling efficient, robust simulations even with moving geometries.

Contribution

It develops a novel hybrid scheme combining asymptotic approximations with boundary integral operators, improving accuracy and efficiency over traditional quadrature methods.

Findings

The method avoids geometrically-induced stiffness.

It utilizes a fast Gauss transform for acceleration.

The approach is adaptable to moving geometries and extendable to three dimensions.

Abstract

We present a hybrid asymptotic/numerical method for the accurate computation of single and double layer heat potentials in two dimensions. It has been shown in previous work that simple quadrature schemes suffer from a phenomenon called "geometrically-induced stiffness," meaning that formally high-order accurate methods require excessively small time steps before the rapid convergence rate is observed. This can be overcome by analytic integration in time, requiring the evaluation of a collection of spatial boundary integral operators with non-physical, weakly singular kernels. In our hybrid scheme, we combine a local asymptotic approximation with the evaluation of a few boundary integral operators involving only Gaussian kernels, which are easily accelerated by a new version of the fast Gauss transform. This new scheme is robust, avoids geometrically-induced stiffness, and is easy to use in the presence of moving geometries. Its extension to three dimensions is natural and straightforward, and should permit layer heat potentials to become flexible and powerful tools for modeling diffusion processes.