Transport-collapse scheme for scalar conservation laws -- initial and boundary value problems

TL;DR

This paper extends Brenier's transport collapse scheme to heterogeneous scalar conservation laws and initial-boundary value problems, providing a new numerical scheme and solution concept with demonstrated numerical examples.

Contribution

It introduces a generalized transport collapse scheme for scalar conservation laws, applicable to initial-boundary value problems, and proposes a new solution concept with numerical validation.

Findings

Scheme converges to entropy solutions

New solution concept for initial-boundary value problems

Numerical examples demonstrate effectiveness

Abstract

We extend Brenier's transport collapse scheme on the Cauchy problem for heterogeneous scalar conservation laws and initial-boundary value problem for homogeneous scalar conservation laws. It is based on averaging out the solution to the corresponding kinetic equation, and it necessarily converges toward the entropy admissible solution. In the case of initial-boundary value problem, we such a procedure is used to construct a numerical scheme which leads us to a new solution concept for initial-boundary value problem for scalar conservation laws. The concept is a generalization (refinement) of the previous works on initial-boundary value problem. We also provide numerical examples.