Analysis and numerical approximation of conservation laws with discontinuous

Abstract

This thesis deals with an analytical as well as numerical study of conservation laws with discontinuous coefficients. We restrict ourselves to the case of single conservation laws and in one space dimension. The conservation laws have spatially varying and possibly discontinuous coefficients. Such equations frequently arise in various physical models, like in two-phase flows in a heterogeneous porous medium and in modeling the action of the ideal clarifier-thickener units used in wastewater treatment plants.

The aim of the thesis is the development of a suitable entropy framework for these equations and the design and analysis of efficient numerical methods to approximate the solutions. In this direction, we have proposed an alternative entropy framework that is based on a two-step approach. In the first step, interface connections are identified and entropy solutions are characterized in terms of them. We show ?? 1 L 1 stability of the entropy solutions with respect to each connection. This step allows us to incorporate different solution concepts in a single framework.

In the second step, a cost functional is defined at the interface, and its minimizers are singled out as the optimal entropy solutions. These optimal entropy solutions, in the case of convex type fluxes, are consistent with the expected solutions for two-phase flows in a porous medium. The existence of solutions is shown by designing suitable Godunov-type schemes and showing that they converge to the entropy solutions. These schemes are easy to implement on account of explicit formulas for the interface fluxes. The convergence of the schemes for complicated flux geometries involves non-trivial modifications of the singular mapping technique. Numerical experiments are presented to compare the performance of these schemes with other existing methods.