An integral equation method for the advection-diffusion equation on time-dependent domains in the plane

TL;DR

This paper introduces an integral equation-based numerical method for solving the advection-diffusion equation on moving and deforming two-dimensional geometries, combining high-order time-stepping and specialized quadrature techniques.

Contribution

It develops a novel integral equation approach with adaptive semi-implicit spectral deferred correction for time integration on time-dependent domains.

Findings

Method achieves high accuracy and robustness.

Handles complex, deforming geometries effectively.

Demonstrates computational efficiency with elliptic marching.

Abstract

Boundary integral methods are attractive for solving homogeneous linear constant coefficient elliptic partial differential equations on complex geometries, since they can offer accurate solutions with a computational cost that is linear or close to linear in the number of discretization points on the boundary of the domain. However, these numerical methods are not straightforward to apply to time-dependent equations, which often arise in science and engineering. We address this problem with an integral equation-based solver for the advection-diffusion equation on moving and deforming geometries in two space dimensions. In this method, an adaptive high-order accurate time-stepping scheme based on semi-implicit spectral deferred correction is applied. One time-step then involves solving a sequence of non-homogeneous modified Helmholtz equations, a method known as elliptic marching. Our solution methodology utilizes several recently developed methods, including special purpose quadrature, a function extension technique and a spectral Ewald method for the modified Helmholtz kernel. Special care is also taken to handle the time-dependent geometries. The numerical method is tested through several numerical examples to demonstrate robustness, flexibility and accuracy