Positivity--preserving numerical scheme for hyperbolic systems with $\delta\,-$ shock solutions and its convergence analysis

TL;DR

This paper introduces a positivity-preserving numerical scheme for hyperbolic systems with delta-shock solutions, ensuring convergence and physical property preservation, and extends it to second-order accuracy with slope limiters.

Contribution

It develops a novel Godunov-type scheme that handles non-classical delta-shocks and maintains positivity and velocity bounds, with proven convergence and improved accuracy.

Findings

The scheme converges to the true solution.

It effectively captures delta-shock solutions.

The extended scheme is second-order accurate.

Abstract

Godunov type numerical schemes for the class of hyperbolic systems, admitting non-classical $\delta-$ shocks are proposed. It is shown that the numerical approximations converge to the solution and preserve the physical properties of the system such as positive density and bounded velocity. The scheme has been extended to positivity preserving and velocity bound preserving second-order accurate scheme by using appropriate slope limiters. The numerical results are compared with the existing the literature and the scheme is shown to capture the solution efficiently. The paper presents a hyperbolic system, for which an entropy satisfying scheme is constructed through an appropriate decoupling of the system into two scalar conservation laws with discontinuous flux.