American Institute of Mathematical Sciences

Advanced
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 - A, 2000, 6 (2) : 419-430. doi: 10.3934/dcds.2000.6.419
[1]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075
[2]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323
[3]

Wei-Chieh Chen, Bogdan Kazmierczak. Traveling waves in quadratic autocatalytic systems with complexing agent. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020364
[4]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299
[5]

Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561
[6]

Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328
[7]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256
[8]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003
[9]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215
[10]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466
[11]

Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021008
[12]

Yubiao Liu, Chunguo Zhang, Tehuan Chen. Stabilization of 2-d Mindlin-Timoshenko plates with localized acoustic boundary feedback. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021006
[13]

Gui-Qiang Chen, Beixiang Fang. Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 85-114. doi: 10.3934/dcds.2009.23.85
[14]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185
[15]

Tianwen Luo, Tao Tao, Liqun Zhang. Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3737-3765. doi: 10.3934/dcds.2019230
[16]

Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020408
[17]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050
[18]

Julian Koellermeier, Giovanni Samaey. Projective integration schemes for hyperbolic moment equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021008
[19]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168
[20]

Yuxi Zheng. Absorption of characteristics by sonic curve of the two-dimensional Euler equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 605-616. doi: 10.3934/dcds.2009.23.605

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (90)
  • HTML views (0)
  • Cited by (21)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences