The lightning method for the heat equation

TL;DR

This paper presents a novel numerical method combining the Lightning Method and Laplace transform inversion to solve the heat equation with spectral accuracy and robustness, especially in domains with sharp corners and solution singularities.

Contribution

It introduces a new approach that integrates the Lightning Method with Laplace transform techniques for efficient, accurate solutions of the heat equation in complex geometries.

Findings

Achieves spectral accuracy with root-exponential convergence.

Robust performance across various time intervals.

Effective handling of domains with sharp corners.

Abstract

This paper introduces a new method for solving the planar heat equation based on the Lightning Method. The lightning method is a recent development in the numerical solution of linear PDEs which expresses solutions using sums of polynomials and rational functions, or more generally as sums of fundamental solutions. The method is particularly well suited to handle domains with sharp corners where solution singularities are present. Boundary conditions are formed on a set of collocation points which is then solved as an overdetermined linear system. The approach of the present work is to utilize the Laplace transform to obtain a modified Helmholtz equation which is solved by an application of the lightning method. The numerical inversion of the Laplace transform is then performed by means of Talbot integration. Our validation of the method against existing results and multiple challenging test problems shows the method attains spectral accuracy with root-exponential convergence while being robust across a wide range of time intervals and adaptable to a variety of geometric scenarios.