A Numerical Study of the Interaction of Waves for 2D Riemann Problem

Abstract

This study is devoted to numerical study of the Riemann problem for scalar conservation equation having two discontinuity surfaces in initial function. It is known that one of the main properties of mentioned problem is that, in the solution of the problem have discontinuity surfaces whose locations are unknown beforehand, even if the initial function is sufficiently smooth. The existence of discontinuity surfaces arises difficulties to find of physical genuine solution by using classical methods. Our aim is to examine the interactions of the waves emerging in solution of the problem in which initial profile having discontinuity surfaces. To this end, we introduce an auxiliary problem that has advantages over the main problem, and using these advantages, an original finite difference method to solve of the auxiliary problem is d eveloped. The solution of the auxiliary problem is used to find the weak solution of the main p roblem. In addition, the auxiliary problem allows writing a higher order finite difference scheme with respect to the time variable. Using the proposed finite difference scheme, computer tests were carried out and the interaction dynamics of shock and rarefaction waves that emerged in the solution of the problem were investigated. © 2023 American Institute of Physics Inc.. All rights reserved.