A note on the Riemann solutions to the isentropic Euler equations in the vanishing pressure limit

TL;DR

This paper investigates the behavior of Riemann solutions to the isentropic Euler equations as the pressure approaches zero, revealing how wave patterns transition between shocks and rarefactions.

Contribution

It provides a detailed analysis of the limiting behavior of Riemann solutions in the vanishing pressure limit, including numerical illustrations.

Findings

Solutions with a 1-shock and 2-rarefaction tend to a two-shock as pressure vanishes.

Solutions with a 1-rarefaction and 2-shock tend to a two-rarefaction.

Numerical tests confirm the theoretical wave pattern transitions.

Abstract

The behaviour of the solutions to the Riemann problem for the isentropic Euler equations when the pressure vanishes is analysed. It is shown that any solution composed of a 1-shock wave and a 2-rarefaction wave tends to a two-shock wave when the pressure gets smaller than a fixed value determined by the Riemann data; by contrast, any solution composed of a 1-rarefaction wave and a 2-shock wave tends to a two-rarefaction wave. The two situations are illustrated with numerical tests.