Decomposition of the Wave Manifold into Lax Admissible Regions and its Application to the Solution of Riemann Problems

TL;DR

This paper introduces a three-dimensional wave manifold that decomposes into Lax admissible regions, providing a novel method for solving Riemann problems in systems with quadratic flux functions, including non-strictly hyperbolic cases.

Contribution

It develops a wave manifold framework that classifies shock waves and regularizes solutions for Riemann problems with non-strict hyperbolicity and elliptic regions.

Findings

Successfully solves Riemann problems using the wave manifold approach.

Demonstrates continuity of solutions with respect to initial data.

Regularizes solutions despite elliptic regions.

Abstract

We utilize a three-dimensional manifold to solve Riemann Problems that arise from a system of two conservation laws with quadratic flux functions. Points in this manifold represent potential shock waves, hence its name wave manifold. This manifold is subdivided into regions according to the Lax admissibility inequalities for shocks. Finally, we present solutions for the Riemann Problems for various cases and exhibit continuity relative to $L$ and $R$ data, despite the fact that the system is not strictly hyperbolic. The usage of this manifold regularizes the solutions despite the presence of an elliptic region.