American Institute of Mathematical Sciences

Advanced
October  1998, 4(4): 609-634. doi: 10.3934/dcds.1998.4.609

On two-dimensional Riemann problem for pressure-gradient equations of the Euler system

Peng Zhang 1, , Jiequan Li 2, and  Tong Zhang 3,

1. 

Beijing Information and Technology Institute, Beijing, 100101, China
2. 

Institute of Applied Mathematics, Academia Sinica, Beijing, 100080, China
3. 

Institute of Mathematics, Academia Sinica, Beijing, 100080

Received  December 1997 Published  July 1998

We consider the two-dimensional Riemann problem for the pressure-gradient equations with four pieces of initial data, so restricted that only one elementary wave appears at each interface. This model comes from the flux-splitting of the compressible Euler system. Lack of the velocity in the eigenvalues, the slip lines have little influence on the structures of solutions. The flow exhibits the simpler patterns than in the Euler system, which makes it possible to clarify the interaction of waves in two dimensions. The present paper is devoted to analyzing the structures of solutions and presenting numerical results to the two-dimensional Riemann problem. Especially, we give the criterion of transition from the regular reflection to the Mach reflection in the interaction of shocks.
Keywords: two-dimensional Riemann problem, regular reflection, Mach reflection, Pressure-gradient equations, rarefaction wave, slip line, MmB scheme., shock.
Mathematics Subject Classification: 35L65, 58F40, 65M06, 76N1.
Citation: Peng Zhang, Jiequan Li, Tong Zhang. On two-dimensional Riemann problem for pressure-gradient equations of the Euler system. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 609-634. doi: 10.3934/dcds.1998.4.609
[1]

Hanchun Yang, Meimei Zhang, Qin Wang. Global solutions of shock reflection problem for the pressure gradient system. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3387-3428. doi: 10.3934/cpaa.2020150
[2]

Eun Heui Kim, Charis Tsikkou. Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6257-6289. doi: 10.3934/dcds.2017271
[3]

Lihui Guo, Wancheng Sheng, Tong Zhang. The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system$^*$. Communications on Pure & Applied Analysis, 2010, 9 (2) : 431-458. doi: 10.3934/cpaa.2010.9.431
[4]

Jerzy Gawinecki, Wojciech M. Zajączkowski. Global regular solutions to two-dimensional thermoviscoelasticity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1009-1028. doi: 10.3934/cpaa.2016.15.1009
[5]

Qin Wang, Kyungwoo Song. The regularity of sonic curves for the two-dimensional Riemann problems of the nonlinear wave system of Chaplygin gas. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1661-1675. doi: 10.3934/dcds.2016.36.1661
[6]

Jiequan Li, Mária Lukáčová - MedviĎová, Gerald Warnecke. Evolution Galerkin schemes applied to two-dimensional Riemann problems for the wave equation system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 559-576. doi: 10.3934/dcds.2003.9.559
[7]

Eemeli Blåsten, Oleg Yu. Imanuvilov, Masahiro Yamamoto. Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials. Inverse Problems & Imaging, 2015, 9 (3) : 709-723. doi: 10.3934/ipi.2015.9.709
[8]

Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985
[9]

Weinan E, Jianchun Wang. A thermodynamic study of the two-dimensional pressure-driven channel flow. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4349-4366. doi: 10.3934/dcds.2016.36.4349
[10]

Hélène Hibon, Ying Hu, Yiqing Lin, Peng Luo, Falei Wang. Quadratic BSDEs with mean reflection. Mathematical Control & Related Fields, 2018, 8 (3&4) : 721-738. doi: 10.3934/mcrf.2018031
[11]

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
[12]

Faustino Maestre, Pablo Pedregal. Dynamic materials for an optimal design problem under the two-dimensional wave equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 973-990. doi: 10.3934/dcds.2009.23.973
[13]

Nguyen Huy Tuan, Tran Ngoc Thach, Yong Zhou. On a backward problem for two-dimensional time fractional wave equation with discrete random data. Evolution Equations & Control Theory, 2020, 9 (2) : 561-579. doi: 10.3934/eect.2020024
[14]

Sergey A. Denisov. Infinite superlinear growth of the gradient for the two-dimensional Euler equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 755-764. doi: 10.3934/dcds.2009.23.755
[15]

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
[16]

Dongfen Bian. Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1591-1611. doi: 10.3934/dcdss.2016065
[17]

Eddye Bustamante, José Jiménez Urrea, Jorge Mejía. The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1177-1203. doi: 10.3934/cpaa.2019057
[18]

Pedro Duarte, José Pedro GaivÃo, Mohammad Soufi. Hyperbolic billiards on polytopes with contracting reflection laws. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3079-3109. doi: 10.3934/dcds.2017132
[19]

Terasan Niyomsataya, Ali Miri, Monica Nevins. Decoding affine reflection group codes with trellises. Advances in Mathematics of Communications, 2012, 6 (4) : 385-400. doi: 10.3934/amc.2012.6.385
[20]

Al-hassem Nayam. Constant in two-dimensional $p$-compliance-network problem. Networks & Heterogeneous Media, 2014, 9 (1) : 161-168. doi: 10.3934/nhm.2014.9.161

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (18)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences