Numerical Methods for Conservation Laws with a Discontinuous Flux Function

Abstract

When simulating two-phase flow in a porous medium, numerical methods are used to solve the equations of flow, called conservation laws. In the industry, this is done by a reservoir simulator, and the most widely used method is the Upstream Mobility scheme. It is useful to compare how this scheme solves the flow problem against academically accepted schemes, like Godunov's method and Engquist-Osher's method. To gain knowledge on the numerical approximations, the theory behind must be known, especially when dealing with spatial discontinuities. Only then will a comparision between numerical results be applicable for physical models. In this thesis we have investigated the theory of conservation laws with discontinuous flux functions, introduced a new scheme for this problem, Local Lax-Friedrichs, and compared the Upstream Mobility scheme against the Godunov, Engquist-Osher and Local Lax-Friedrichs scheme.