Approximation of the Interactions of Rarefaction Waves by the Wave Front Tracking Method

Abstract

The interaction of two simple delta shock waves for a pressureless gas dynamic system is considered. The result of the interaction is a delta shock wave with constant speed. This interaction is approximated by letting the perturbed parameter in the Euler equations for isentropic fluids go to zero. Each delta shock wave is approximated by two shock waves of the first and second family when the perturbed parameter goes to zero. These shock waves are solutions of two Riemann problems at time t=0. The solution of the Riemann problem for t>0 can also contain rarefaction waves. If the perturbed parameter approaches 0, the strength of the rarefaction waves increases and the number of interactions of the rarefaction waves increases, as well. When two split rarefaction waves interact, the number of Riemann problems to be solved is m1·m2, where mi is the number of ith rarefaction waves. The main topic of this paper is to develop an algorithm that reduces the number of these Riemann problems. The algorithm is based on the determination of the intermediate states that make the Rankine–Hugoniot deficit small. The approximated wave front tracking algorithm was used for the numerical verification of these interactions. The theoretical background was the concept of the shadow wave solution.