Revisiting the method of characteristics via a convex hull algorithm

TL;DR

This paper introduces a convex hull algorithm (CHA) for solving nonlinear hyperbolic problems, enabling computation of entropy solutions and handling shock formation effectively across various equations.

Contribution

The paper presents a novel convex hull algorithm that computes entropy dissipative and multivalued solutions for hyperbolic conservation laws, applicable to convex and non-convex cases.

Findings

Successfully computes entropy solutions after shock formation.

Applicable to Hamilton-Jacobi equations and other characteristic problems.

Demonstrated effectiveness through numerical tests including fluid dynamics.

Abstract

We revisit the method of characteristics for shock wave solutions to nonlinear hyperbolic problems and we describe a novel numerical algorithm - the convex hull algorithm (CHA) - in order to compute, both, entropy dissipative solutions (satisfying all relevant entropy inequalities) and entropy conservative (or multivalued) solutions to nonlinear hyperbolic conservation laws. Our method also applies to Hamilton-Jacobi equations and other problems endowed with a method of characteristics. From the multivalued solutions determined by the method of characteristic, our algorithm "extracts" the entropy dissipative solutions, even after the formation of shocks. It applies to, both, convex or non-convex flux/Hamiltonians. We demonstrate the relevance of the proposed approach with a variety of numerical tests including a problem from fluid dynamics.