On convergence of explicit finite volume scheme for one-dimensional three-component two-phase flow model in porous media

Abstract

In this work, we develop and analyze an explicit finite volume scheme for a one-dimensional nonlinear, degenerate, convection–diffusion equation having application in petroleum reservoir. The main difficulty is that the solution typically lacks regularity due to the degenerate nonlinear diffusion term. We analyze a numerical scheme corresponding to explicit discretization of the diffusion term and a Godunov scheme for the advection term. L∞{L}^{\infty } stability under appropriate CFL conditions and BV{\rm{BV}} estimates are obtained. It is shown that the scheme satisfies a discrete maximum principle. Then we prove convergence of the approximate solution to the weak solution of the problem, and we mount convergence results to a weak solution of the problem in L1{L}^{1}. Results of numerical experiments are presented to validate the theoretical analysis.

Keywords

• finite volume method
• degenerate parabolic equation
• nonlinear convection–diffusion
• porous media
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