A Green's Function Numerical Method for Solving Parabolic Partial Differential Equations

Abstract

This article describes the derivation and implementation of a numerical method to solve constant-coecient, parabolic partial dierential equations in two space dimensions on rect-angular domains. The method is based on a formula for the Green's function for the problem obtained via re ections at the boundary of the domain from the corresponding formula for the fundamental solution in the whole plane. It is inspired by a related method for variable coe-cients equations in the whole space introduced by Constantinescu, Costanzino, Mazzucato, and Nistor inJ. Math. Phys , 51 103502 (2010). The benchmark case of the two-dimensional heat equation is considered. We compare the Green's function method with a nite-dierence scheme, more precisely, an alternating direction implicit (ADI) method due to Peaceman and Rachford. Our method yields better rates of convergence to the exact solution.