Formation of vacuum state and delta-shock in the solution of two-dimensional Riemann problem for zero pressure gas dynamics

TL;DR

This paper analyzes the formation of delta shocks and vacuum regions in two-dimensional zero-pressure gas dynamics, revealing nine distinct wave interaction patterns through theoretical analysis and numerical simulations.

Contribution

It introduces a comprehensive classification of solution patterns involving delta shocks and vacuum formation in 2D pressureless Euler equations, supported by analytical and numerical validation.

Findings

Nine topologically distinct solution patterns identified.

Delta shocks characterized by Dirac delta functions in density and energy.

Numerical simulations show excellent agreement with theoretical predictions.

Abstract

In this article, we investigate the two-dimensional pressureless Euler equations with three constant Riemann initial data. Our primary focus is on the wave interactions involving contact discontinuities and delta shocks. A distinguishing feature of the solution is the emergence of a delta shock wave which is characterized by a Dirac delta function appearing in both the density and internal energy variables. By exploiting generalized characteristic analysis, nine topologically distinct solution patterns are derived. Some of these configurations exhibit features similar to Mach-reflection and in certain cases, vacuum regions may also develop. To validate the theoretical results, numerical simulations are carried out using a semidiscrete central upwind scheme. The comparison between analytical and numerical results demonstrates excellent agreement, providing