A numerical scheme for singular shock solutions and a study of its consistence in the sense of distributions

TL;DR

This paper introduces a numerical scheme for approximating singular shock solutions in the Keyfitz-Kranzer system, demonstrating its consistency in the distributional sense and applicability to degenerate systems through specific examples.

Contribution

The paper develops and proves the distributional consistency of a numerical scheme for singular shocks, including degenerate systems, with detailed analysis and examples.

Findings

Scheme is consistent in the sense of distributions as space step tends to zero.

The scheme adapts effectively to degenerate systems.

Full proof of consistency for the Korchinski model is provided.

Abstract

In this paper we present a numerical scheme for the approximation of singular shock solutions of the Keyfitz-Kranzer model system. Consistence in the sense of distributions is studied. As long as some numerical properties are verified when the space step tends to 0, we prove that the scheme provides a numerical solution that satisfies the equations in the sense of distributions with an approximation that tends to 0 when h \rightarrow 0. We also show that this scheme adapts to degenerate systems. This is illustrated by two examples: the system presenting delta wave solutions originally studied by Korchinski and another system studied by Keyfitz-Kranzer that models elasticity. Consistence of the scheme in the sense of distributions is fully proved in the case of the Korchinski model.