A Finite Difference Method for Two-PhaseParabolic Obstacle-like Problem

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A Finite Difference Method for Two-PhaseParabolic Obstacle-like Problem Avetik Arakelyan

In this paper we treat the numerical approximationof the two-phase parabolic obstacle-like problem:∆u−ut=λ+·χ{u>0}−λ−·χ{u<0},(t,x)∈(0,T)×Ω,whereT <∞,λ+,λ−>0 are Lipschitz continuous functions,and Ω⊂Rnis a bounded domain. We introduce a certainvariation form, which allows us to define a notion of viscositysolution. We use defined viscosity solutions framework to ap-ply Barles-Souganidis theory. The numerical projected Gauss-Seidel method is constructed. Although the paper is devotedto the parabolic version of the two-phase obstacle-like problem,we prove convergence of the discretized scheme to the uniqueviscosity solution for both two-phase parabolic obstacle-like andstandard two-phase membrane problem. Numerical simulationsare also presented.

DOI: https://armjmath.sci.am/index.php/ajm/article/view/109 Armenian Journal of Mathematics, 7(1) 32-49

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Keywords

Free boundary, Two-phase obstacle-like equation, Finite differ-ence, Viscosity solution

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