A Finite Difference Method for Two-Phase Parabolic Obstacle-like Problem

TL;DR

This paper develops a finite difference numerical scheme for a two-phase parabolic obstacle problem, proving convergence to the unique viscosity solution and demonstrating its effectiveness through numerical simulations.

Contribution

It introduces a variational form and viscosity solutions framework for the problem, and constructs a convergent numerical projected Gauss-Seidel method.

Findings

Proved convergence of the scheme to the viscosity solution.

Validated the method with numerical simulations.

Applicable to both parabolic obstacle and membrane problems.

Abstract

In this paper we treat the numerical approximation of the two-phase parabolic obstacle-like problem: \[\Delta u -u_t=\lambda^+\cdot\chi_{\{u>0\}}-\lambda^-\cdot\chi_{\{u<0\}},\quad (t,x)\in (0,T)\times\Omega,\] where $T < \infty, \lambda^+ ,\lambda^- > 0$ are Lipschitz continuous functions, and $\Omega\subset\mathbb{R}^n$ is a bounded domain. We introduce a certain variational form, which allows us to define a notion of viscosity solution. We use defined viscosity solutions framework to apply Barles-Souganidis theory. The numerical projected Gauss-Seidel method is constructed. Although the paper is devoted to the parabolic version of the two-phase obstacle-like problem, we prove convergence of the discretized scheme to the unique viscosity solution for both two-phase parabolic obstacle-like and standard two-phase membrane problem. Numerical simulations are also presented.