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

Abstract

In this paper we treat the numerical approximation of the two-phase parabolic obstacle-like problem: \[Δu -u_t=λ^+\cdotχ_{\{u>0\}}-λ^-\cdotχ_{\{u<0\}},\quad (t,x)\in (0,T)\timesΩ,\] where $T < \infty, λ^+ ,λ^- > 0$ are Lipschitz continuous functions, and $Ω\subset\mathbb{R}^n$ 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.