Global Riemann Solvers for Several $3\times3$ Systems of Conservation Laws with Degeneracies

TL;DR

This paper develops global Riemann solvers for complex 3x3 conservation law systems with degeneracies, relevant to modeling two-phase flow and traffic flow in heterogeneous media, by exploiting their structural properties.

Contribution

It introduces novel methods to construct Riemann solvers for degenerate 3x3 systems, including partially decoupled and reduced models, advancing the analysis of such complex systems.

Findings

Constructed global Riemann solvers for multiple 3x3 systems

Identified structural features enabling decoupling and reduction

Discussed approaches for solving associated Cauchy problems

Abstract

We study several $3\times 3$ systems of conservation laws, arising in modeling of two phase flow with rough porous media and traffic flow with rough road condition. These systems share several features. The systems are of mixed type, with various degeneracies. Some families are linearly degenerate, while others are not genuinely nonlinear. Furthermore, along certain curves in the domain the eigenvalues and eigenvectors of different families coincide. Most interestingly, in some suitable Lagrangian coordinate the systems are partially decoupled, where some unknowns can be solved independently of the others. Finally, in special cases the systems reduce to some $2\times2$ models, which have been studied in the literature. Utilizing the insights gained from these features, we construct global Riemann solvers for all these models. Possible treatments on the Cauchy problems are also discussed.