A Relaxation/Finite Difference discretization of a 2D Semilinear Heat Equation over a rectangular domain

TL;DR

This paper introduces a new discretization method combining relaxation and finite differences for a 2D semilinear heat equation, providing unconditionally well-posedness and optimal second order convergence.

Contribution

It adapts a relaxation scheme originally for nonlinear Schrödinger equations to a 2D heat equation, establishing convergence and error estimates.

Findings

Method is unconditionally well-posed.

Achieves optimal second order error convergence.

Works under mild mesh conditions.

Abstract

We consider an initial and Dirichlet boundary value problem for a semilinear, two dimensional heat equation over a rectangular domain. The problem is discretized in time by a version of the Relaxation Scheme proposed by C. Besse (C. R. Acad. Sci. Paris S\'er. I, vol. 326 (1998)) for the nonlinear Schr\"odinger equation and in space by a standard second order finite difference method. The proposed method is unconditionally well-posed and its convergence is established by proving an optimal second order error estimate allowing a mild mesh condition to hold.