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 reso¬lution. 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 approxima¬tion 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 diffu¬sion 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.