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April  2000, 6(2): 419-430. doi: 10.3934/dcds.2000.6.419

On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities

Tung Chang 1, , Gui-Qiang Chen 2, and  Shuli Yang 3,

1. 

Institute of Mathematics, Academia Sinica, Beijing 100080, China
2. 

Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730
3. 

Institute of Applied Mathematics, Academia Sinica, Beijing 100080, China

Received  April 1999 Revised  October 1999 Published  January 2000

We are concerned with the Riemann problem for the two-dimensional compressible Euler equations in gas dynamics. This paper is a continuation of our program (see [CY1,CY2]) in studying the interaction of nonlinear waves in the Riemann problem. The central point in this issue is the dynamical interaction of shock waves, centered rarefaction waves, and contact discontinuities that connect two neighboring constant initial states in the quadrants. In this paper we focus mainly on the interaction of contact discontinuities, which consists of two genuinely different cases. For each case, the structure of the Riemann solution is analyzed by using the method of characteristics, and the corresponding numerical solution is illustrated via contour plots by using the upwind averaging scheme that is second-order in the smooth region of the solution developed in [CY1]. For one case, the four contact discontinuities role up and generate a vortex, and the density monotonically decreases to zero at the center of the vortex along the stream curves. For the other, two shock waves are formed and, in the subsonic region between two shock waves, a new kind of nonlinear hyperbolic waves (called smoothed Delta-shock waves) is observed.
Keywords: rarefaction waves, 2-D Riemann problem, shock waves, interaction of waves, vortices, contact discontinuities, upwind averaging schemes., transonic flow, Euler equations.
Mathematics Subject Classification: Primary: 76N10, 35L65, 35L67; Secondary: 65M0.
Citation: Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 419-430. doi: 10.3934/dcds.2000.6.419
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