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

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-infinity stability under appropriate CFL conditions and 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 L-1. Results of numerical experiments are presented to validate the theoretical analysis.