Implicit monotone difference methods for scalar conservation laws with source terms

TL;DR

This paper introduces new implicit monotone difference methods for scalar conservation laws with source terms, extending previous work, establishing a connection to entropy inequalities, and providing convergence proofs validated by numerical tests.

Contribution

It develops a novel implicit framework centered on monotonicity for scalar conservation laws with source terms, including a new convergence proof not relying on classical compactness.

Findings

Established a link between numerical schemes and discrete entropy inequalities

Developed three implicit methods based on the new notions

Confirmed theoretical results through numerical experiments

Abstract

In this article, a concept of implicit methods for scalar conservation laws in one or more spatial dimensions allowing also for source terms of various types is presented. This material is a significant extension of previous work of the first author [3]. Implicit notions are developed that are centered around a monotonicity criterion. We demonstrate a connection between a numerical scheme and a discrete entropy inequality, which is based on a classical approach by Crandall and Majda. Additionally, three implicit methods are investigated using the developed notions. Next, we conduct a convergence proof which is not based on a classical compactness argument. Finally, the theoretical results are confirmed by various numerical tests.