Stability of Riemann solutions to pressureless Euler equations with Coulomb-like friction by flux approximation

TL;DR

This paper proves that Riemann solutions to pressureless Euler equations with Coulomb-like friction remain stable under flux approximation, with solutions of the approximated system converging to the original as the perturbation parameter approaches zero.

Contribution

It extends the stability analysis of Riemann solutions from homogeneous to nonhomogeneous pressureless Euler equations with Coulomb-like friction under flux approximation.

Findings

Riemann solutions of the approximated system converge to the original solutions as perturbation tends to zero.

The approximated system is strictly hyperbolic with distinct eigenvalues, differing from the original system.

The stability result generalizes previous work from homogeneous to nonhomogeneous cases.

Abstract

We study the stability of Riemann solutions to pressureless Euler equations with Coulomb-like friction under the nonlinear approximation of flux functions with one parameter. The approximated system can be seen as the generalized Chaplygin pressure Aw-Rascle model with Coulomb-like friction, which is also equivalent to the nonsymmetric system of Keyfitz-Kranzer type with generalized Chaplygin pressure and Coulomb-like friction. Compared with the original system, The approximated system is strictly hyperbolic, which has one eigenvalue genuinely nonlinear and the other linearly degenerate. Hence, the structure of its Riemann solutions is much different from the ones of the original system. However, it is proven that the Riemann solutions for the approximated system converge to the corresponding ones to the original system as the perturbation parameter tends to zero, which shows that the Riemann solutions to nonhomogeneous pressureless Euler equations is stable under such kind of flux approximation. The result in this paper generalizes the stability of Riemann solutions with respect to flux perturbation from the well-known homogeneous case to the nonhomogeneous case.