Fast integral equation methods for linear and semilinear heat equations in moving domains

TL;DR

This paper introduces integral equation-based solvers for heat equations in complex moving domains, offering improved accuracy, stability, and adaptivity over traditional methods, supported by fast algorithms and numerical demonstrations.

Contribution

The paper develops a comprehensive framework combining fast heat potential evaluation, high-order quadratures, and adaptive meshing for solving heat equations in dynamic geometries.

Findings

Efficient algorithms for heat potential evaluation

High-order quadratures for singular integrals

Numerical examples demonstrating method performance

Abstract

We present a family of integral equation-based solvers for the linear or semilinear heat equation in complicated moving (or stationary) geometries. This approach has significant advantages over more standard finite element or finite difference methods in terms of accuracy, stability and space-time adaptivity. In order to be practical, however, a number of technical capabilites are required: fast algorithms for the evaluation of heat potentials, high-order accurate quadratures for singular and weakly integrals over space-time domains, and robust automatic mesh refinement and coarsening capabilities. We describe all of these components and illustrate the performance of the approach with numerical examples in two space dimensions.