# Flux-approximation limits of solutions to the Brio system with two independent parameters

## TL;DR

This paper analyzes the limits of Riemann solutions to the Brio system with two parameters using flux-approximation, revealing how solutions converge to delta-shocks, vacuum states, or contact discontinuities as parameters vanish.

## Contribution

It provides a detailed analytical study of the flux-approximation limits of the Brio system, including convergence to delta-shocks and vacuum solutions, and clarifies formation mechanisms.

## Key findings

- Solutions converge to delta-shocks and vacuum states as parameters vanish.
- Two-shock and two-rarefaction solutions tend to delta-shocks and contact discontinuities.
- Mechanisms of delta-shock formation under flux approximation are clarified.

## Abstract

By the flux-approximation method, we study limits of Riemann solutions to the Brio system with two independent parameters. The Riemann problem of the perturbed system is solved analytically, and four kinds of solutions are obtained constructively. It is shown that, as the two-parameter flux perturbation vanishes, any two-shock-wave and two-rarefaction-wave solutions of the perturbed Brio system converge to the delta-shock and vacuum solutions of the transport equations, respectively. In addition, we specially pay attention to the Riemann problem of a simplified system of conservation laws derived from the perturbed Brio system by neglecting some quadratic term. As one of the purebred parameters of the Brio system goes to zero, the solution of which consisting of two shock waves tends to a delta-shock solution to this simplified system. By contrast, the solution containing two rarefaction waves converges to a contact discontinuity and a rarefaction wave of the simplified system. What is more, the formation mechanisms of delta shock waves under flux approximation with both two parameters and only one parameter are clarified.