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

Arakelyan, Avetik (2015) A Finite Difference Method for Two-Phase Parabolic Obstacle-like Problem. Armenian Journal of Mathematics, 7 (1). pp. 32-49. ISSN 1829-1163

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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 variation 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.

Item Type: Article 35-xx Partial differential equations65-xx Numerical analysis 607 Dr Avetik Arakelyan 27 May 2015 00:42 27 May 2015 00:42

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