Backward Euler Approximations for Conservation Laws with Discontinuous Flux

TL;DR

This paper investigates backward Euler methods for solving conservation laws with discontinuous flux, establishing their convergence to entropy solutions using semigroup theory and viscous approximations.

Contribution

It introduces a novel approach to prove convergence of backward Euler schemes for discontinuous flux conservation laws via viscous approximation analysis.

Findings

Existence of backward Euler approximations for the problem

Convergence of these approximations to entropy solutions

Application of semigroup theory to nonlinear conservation laws

Abstract

Solutions to a class of conservation laws with discontinuous flux are constructed relying on the Crandall-Liggett theory of nonlinear contractive semigroups~\cite{CL}. In particular, the paper studies the existence of backward Euler approximations, and their convergence to a unique entropy-admissible solution to the Cauchy problem. The proofs are achieved through the study of the backward Euler approximations to the viscous conservation laws.