Resonance in rarefaction and shock curves: local analysis and numerics of the continuation method

TL;DR

This paper develops a numerical continuation method for solving Riemann problems in systems of conservation laws, addressing challenges posed by loss of hyperbolicity and nonlinearity, with theoretical analysis and practical examples.

Contribution

It introduces a novel continuation procedure for wave curves beyond points of wave speed coincidence, regularizes loss of hyperbolicity, and proves the existence of composite waves in non-genuinely nonlinear systems.

Findings

Successful continuation of wave curves at maximal codimensionality points

Regularization of hyperbolicity loss via Generalized Jordan Chain

Numerical examples demonstrating method effectiveness

Abstract

In this paper, we describe certain crucial steps in the development of an algorithm for finding the Riemann solution in systems of conservation laws. We relax the classical hypotheses of strict hyperbolicity and genuine nonlinearity of Lax. First, we present a procedure for continuing wave curves beyond points where characteristic speeds coincide, i.e., at wave curve points of maximal codimensionality. This procedure requires strict hyperbolicity on both sides of the coincidence locus. Loss of strict hyperbolicity is regularized a Generalized Jordan Chain, which serves to construct a four-fold submanifold structure on which wave curves can be continued. Second, we analyze the case of loss of genuine nonlinearity. We prove a new result: the existence of composite wave curves when the composite wave traverses either the inflection locus or an anomalous part of the non-local composite wave curve. In this sense, we find conditions under which the composite field is well defined and its singularities can be removed, allowing use of our continuation method. Finally, we present numerical examples for a non-strictly hyperbolic system of conservation laws.