# American Institute of Mathematical Sciences

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

 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.
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 - A, 2000, 6 (2) : 419-430. doi: 10.3934/dcds.2000.6.419
 [1] 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 - A, 1995, 1 (4) : 555-584. doi: 10.3934/dcds.1995.1.555 [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 [3] Hualin Zheng. Stability of a superposition of shock waves with contact discontinuities for the Jin-Xin relaxation system. Kinetic & Related Models, 2015, 8 (3) : 559-585. doi: 10.3934/krm.2015.8.559 [4] Min Ding, Hairong Yuan. Stability of transonic jets with strong rarefaction waves for two-dimensional steady compressible Euler system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2911-2943. doi: 10.3934/dcds.2018125 [5] Xiao-Biao Lin, Stephen Schecter. Traveling waves and shock waves. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : i-ii. doi: 10.3934/dcds.2004.10.4i [6] Dongbing Zha. Remarks on nonlinear elastic waves in the radial symmetry in 2-D. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 4051-4062. doi: 10.3934/dcds.2016.36.4051 [7] Feng Xie. Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1075-1100. doi: 10.3934/dcdsb.2012.17.1075 [8] Christian Rohde, Wenjun Wang, Feng Xie. Hyperbolic-hyperbolic relaxation limit for a 1D compressible radiation hydrodynamics model: superposition of rarefaction and contact waves. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2145-2171. doi: 10.3934/cpaa.2013.12.2145 [9] Yanbo Hu, Tong Li. The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3317-3336. doi: 10.3934/cpaa.2019149 [10] James K. Knowles. On shock waves in solids. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 573-580. doi: 10.3934/dcdsb.2007.7.573 [11] José R. Quintero. Nonlinear stability of solitary waves for a 2-d Benney--Luke equation. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 203-218. doi: 10.3934/dcds.2005.13.203 [12] Yuri Gaididei, Anders Rønne Rasmussen, Peter Leth Christiansen, Mads Peter Sørensen. Oscillating nonlinear acoustic shock waves. Evolution Equations & Control Theory, 2016, 5 (3) : 367-381. doi: 10.3934/eect.2016009 [13] Tong Yang, Huijiang Zhao. Asymptotics toward strong rarefaction waves for $2\times 2$ systems of viscous conservation laws. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 251-282. doi: 10.3934/dcds.2005.12.251 [14] Frederike Kissling, Christian Rohde. The computation of nonclassical shock waves with a heterogeneous multiscale method. Networks & Heterogeneous Media, 2010, 5 (3) : 661-674. doi: 10.3934/nhm.2010.5.661 [15] Martina Chirilus-Bruckner, Guido Schneider. Interaction of oscillatory packets of water waves. Conference Publications, 2015, 2015 (special) : 267-275. doi: 10.3934/proc.2015.0267 [16] Masashi Ohnawa. Convergence rates towards the traveling waves for a model system of radiating gas with discontinuities. Kinetic & Related Models, 2012, 5 (4) : 857-872. doi: 10.3934/krm.2012.5.857 [17] Gero Friesecke, Karsten Matthies. Geometric solitary waves in a 2D mass-spring lattice. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 105-144. doi: 10.3934/dcdsb.2003.3.105 [18] Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic & Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004 [19] Michiel Bertsch, Masayasu Mimura, Tohru Wakasa. Modeling contact inhibition of growth: Traveling waves. Networks & Heterogeneous Media, 2013, 8 (1) : 131-147. doi: 10.3934/nhm.2013.8.131 [20] Gui-Qiang G. Chen, Hairong Yuan. Local uniqueness of steady spherical transonic shock-fronts for the three--dimensional full Euler equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2515-2542. doi: 10.3934/cpaa.2013.12.2515

2018 Impact Factor: 1.143