Analysis of upwind and high-resolution schemes for solving convection dominated problems in porous media

Abstract

The conservation laws governing the multiphase flows in porous media are often convection-dominated and have a steep fronts that require accurate resolution. Standard discretization methods of the convection terms do not perform well for such problems. The main aim of this work is to analyze the use of upwind and high- resolution schemes in such cases. First, we use a first differential approximation method to perform a theoretical analysis of a standard upwind approximation and different time stepping schemes for the linear hyperbolic equations in 1- and 2D. Next, we present a popular approach to reduce the amount of numerical diffusion introduced by upwind approximation - high-resolution schemes. We compare our implementation of one of the recently proposed central-upwind schemes against the upwind schemes on several test problems based on Buckley-Leverett equation and discuss the results. Finally, a parallel version of central-upwind scheme in 2D is presented. It was implemented using our C++ library of parallel arrays - ParSol.