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

In this paper we treat the numerical approximation of the two-phase parabolic obstacle-like problem: $$\mathrm{\Delta }u-u_t={\lambda }^{+}\cdot {\chi }_{\{u>0\}}-{\lambda }^{-}\cdot {\chi }_{\{u<0\}},(t,x)\in (0,T)\times \mathrm{\Omega },$$ where $T<\mathrm{\infty },{\lambda }^{+},{\lambda }^{-}>0$ are Lipschitz continuous functions, and $\mathrm{\Omega }\subset {\mathbb{R}}^n$ 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.