Delta shock wave interactions via wave front tracking method

TL;DR

This paper investigates delta shock wave interactions in pressureless gas dynamics using wave front tracking and shadow wave solutions, comparing approaches and analyzing the effects of small pressure perturbations.

Contribution

It introduces a novel comparison between shadow wave solutions and wave front tracking for delta shock interactions, including the impact of small pressure perturbations.

Findings

Delta shock interactions can produce non-constant speed shocks.

The perturbed system with small pressure is strictly hyperbolic and solvable.

Numerical results converge to shadow wave solutions as pressure vanishes.

Abstract

In this paper we discuss delta shock interaction problem for a pressureless gas dynamics system with two different ways of approaching the subject. The first one is by using shadow wave solution concept. The result of two delta shock interactions is delta shock with non-constant speed in a general case. The second one is by perturbing the system with a small pressure term. The obtained perturbed system is strictly hyperbolic and its Riemann problem is solvable. We compare a limit of a numerical wave front tracking results as small pressure term vanishes with the shadow wave solution. Key words: weighted shadow waves, delta shock waves, wave front tracking, Riemann problem, interactions