A Finite Difference Method for Two-Phase Parabolic Obstacle-like Problem
This work provides a numerical framework for a class of parabolic free boundary problems, but the contribution is incremental as it extends existing techniques to a specific problem variant.
The paper develops a finite difference method for the two-phase parabolic obstacle-like problem and proves convergence of the discretized scheme to the unique viscosity solution. Numerical simulations demonstrate the method's effectiveness.
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.