Fast integral equation methods for the heat equation and the modified Helmholtz equation in two dimensions

TL;DR

This paper introduces a fast integral equation method for solving the heat and modified Helmholtz equations in two dimensions, utilizing time discretization and fast multipole acceleration to achieve efficient computations.

Contribution

It develops a novel integral equation approach combining time discretization with fast multipole methods for efficient 2D heat and Helmholtz problems.

Findings

Achieves $O(N)$ or $O(N ext{log}N)$ computational cost per time step.

Uses volume and double layer potentials for solution formulation.

Employs fast multipole method for acceleration.

Abstract

We present an efficient integral equation approach to solve the heat equation, $u_t (\x) - \Delta u(\x) = F(\x,t)$, in a two-dimensional, multiply connected domain, and with Dirichlet boundary conditions. Instead of using integral equations based on the heat kernel, we take the approach of discretizing in time, first. This leads to a non-homogeneous modified Helmholtz equation that is solved at each time step. The solution to this equation is formulated as a volume potential plus a double layer potential.The volume potential is evaluated using a fast multipole-accelerated solver. The boundary conditions are then satisfied by solving an integral equation for the homogeneous modified Helmholtz equation. The integral equation solver is also accelerated by the fast multipole method (FMM). For a total of $N$ points in the discretization of the boundary and the domain, the total computational cost per time step is $O(N)$ or $O(N\log N)$.