Riemann problem of Euler equations with singular sources

TL;DR

This paper investigates the Riemann problem for one-dimensional Euler equations with a singular source, revealing new stationary discontinuities and diverse wave structures, and provides a comprehensive classification of solutions.

Contribution

It introduces an eigenvalue-based criterion for stationary discontinuities and characterizes all possible Riemann solution structures with singular sources.

Findings

Exact solutions include stationary discontinuities not present in classical Euler solutions.

Stationary discontinuities are selected using a new monotonicity criterion.

The paper classifies all possible Riemann solution structures with singular sources.

Abstract

This paper is concerned with the Riemann problem of one-dimensional Euler equations with a singular source. The exact solution of this Riemann problem contains a stationary discontinuity induced by the singular source, which is different from all the simple waves in the Riemann solution of classical Euler equations. We propose an eigenvalue-based monotonicity criterion to select the physical curve of this stationary discontinuity. By including this stationary discontinuity as an elementary wave, the structure of Riemann solution becomes diverse, e.g. the number of waves is not fixed and interactions between two waves become possible. Under the double CRP framework, we prove all possible structures of the Riemann solution.