The lightning method for the heat equation

TL;DR

The paper presents a spectral-accurate approach to the planar heat equation in unbounded polygonal domains by marrying the Lightning Method with a Laplace-transform framework and Talbot inversion. The method decomposes the Laplace-domain solution into a Newman (poles) and Runge expansion, with carefully designed pole clustering and collocation near corners to resolve singularities. Numerical inversion via a modified Talbot contour yields robust, high-accuracy time-domain solutions, while validation against boundary-integral, kinetic Monte Carlo, and matched asymptotic methods demonstrates strong agreement across complex geometries and boundary/initial conditions. The work extends the LM’s applicability to parabolic problems, offering a robust tool for problems with corner singularities and multiple absorbing bodies, with potential extensions to Neumann/Robin boundaries and alternative rational-approximation schemes.

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.

Figures (17)

• Figure 1: Schematic of the external geometry $\mathbb{R}^2\setminus\Omega$ on which the parabolic system \ref{['eq:main']} is solved. The set of $N_B$ polygonal bodies is $\Omega := \cup_{k=1}^{N_B}\Omega_{k}$ and $n$ is the outward facing normal vector to $\partial\Omega$.
• Figure 2: Example of Newman pole clustering using the tapered distribution \ref{['eqn:clustering_dist']}, with $\sigma$ in \ref{['eqn:sigma']}.
• Figure 3: The Talbot contour \ref{['eq:Talbot']} and the mid-point quadrature points $\{ s_j\}$. Shown are contours and their discretizations for fixed $t=0.1$ and varying $M$ (left) and varying $t$ with fixed $M=9$ (right).
• Figure 4: Point-wise relative residual and over-sampled relative error for $s=-30.16+8.31$, $m=40$ on panels (a-b): a parametrization of the boundary $\partial \Omega$ starting from $-1-1i$ and going counter-clockwise, and panels (c-d): log-scale on $[0,1]$ of the parametrization of the boundary, showing the sample clustering going deeper than the residual to the corner.
• Figure 5: Panels (a-c): Approximate solution of \ref{['eq:Helmholtz']} for various $s$ in the Talbot contour of $t=0.1$, with $m=90$. $\mathcal{E}^\infty[\hat{u}]$ for each given by (a) $2.02\times 10^{-10}$, (b) $8.24\times 10^{-11}$, (c) $3.58\times 10^{-10}$. (d): Convergence of the relative errors for various $s$ in the Talbot contour for time $t=0.1$ in the same configuration as (a-c).