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October  1995, 1(4): 555-584. doi: 10.3934/dcds.1995.1.555

On the 2-D Riemann problem for the compressible Euler equations I. Interaction of shocks and rarefaction waves

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

1. 

Institute of Mathematics, Academia Sinica, Beijing 100080
2. 

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

Institute of Applied Mathematics, Academia Sinica, Beijing 100080

Received  May 1995 Published  August 1995

We are concerned with the Riemann problem for the two-dimensional compressible Euler equations in gas dynamics. The central point at 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. The Riemann problem is classified into eighteen genuinely different cases. For each configuration, the structure of the Riemann solution is analyzed using the method of characteristics, and corresponding numerical solution is illustrated by contour plots using an upwind averaging scheme that is second order in the smooth region of the solution. In the first paper we mainly focus on the interaction of shocks and rarefaction waves. The theory is developed from an analysis of the structure of the Euler equations and their Riemann solutions in [CC, ZZ] and the MmB scheme [WY].
Keywords: vortices, 2-D Riemann problem, shocks, Euler equations, slip curves, interaction of waves, rarefaction waves, transonic flow, upwind averaging schemes..
Mathematics Subject Classification: Primary: 65M06, 76N10, 35L65, 35L67; Secondary: 65M9.
Citation: Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations I. Interaction of shocks and rarefaction waves. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 555-584. doi: 10.3934/dcds.1995.1.555
[1]

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
[2]

Feimin Huang, Yi Wang, Tong Yang. Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity. Kinetic & Related Models, 2010, 3 (4) : 685-728. doi: 10.3934/krm.2010.3.685
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