scholarly journals The Convergence of Riemann Solutions to the Modified Chaplygin Gas Equations with a Coulomb-Like Friction Term as the Pressure Vanishes

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Yongqiang Fan ◽  
Lihui Guo ◽  
Gan Yin
Keyword(s):  
Source Term ◽  
Chaplygin Gas ◽  
Euler System ◽  
Modified Chaplygin Gas ◽  
Riemann Solutions ◽  
Delta Shock ◽  
Vacuum States ◽  
Friction Term ◽  
Self Similar ◽  
Delta Shock Waves

This paper studies the convergence of Riemann solutions to the inhomogeneous modified Chaplygin gas equations as the pressure vanishes. The delta shock waves and vacuum states occur as the pressure vanishes. The Riemann solutions of inhomogeneous modified Chaplygin gas equations are no longer self-similar. It is obviously different from the Riemann solutions of homogeneous modified Chaplygin gas equations. When the pressure vanishes, the Riemann solutions of the modified Chaplygin gas equations with a coulomb-like friction term converge to the Riemann solutions of the pressureless Euler system with a source term.

scholarly journals Limits of Riemann Solutions to the Relativistic Euler Systems for Chaplygin Gas as Pressure Vanishes

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Gan Yin ◽  
Kyungwoo Song
Keyword(s):  
Shock Wave ◽  
Rarefaction Wave ◽  
Wave Solution ◽  
Chaplygin Gas ◽  
Euler System ◽  
Delta Shock Wave ◽  
Riemann Solutions ◽  
Euler Systems ◽  
Delta Shock ◽  
Shock Wave Solution

Vanishing pressure limits of Riemann solutions to relativistic Euler system for Chaplygin gas are identified and analyzed in detail. Unlike the polytropic or barotropic gas case, as the parameter decreases to a critical value, the two-shock solution converges firstly to a delta shock wave solution to the same system. It is shown that, as the parameter decreases, the strength of the delta shock increases. Then as the pressure vanishes ultimately, the solution is nothing but the delta shock wave solution to the zero pressure relativistic Euler system. Meanwhile, the two-rarefaction wave solution and the solution containing one-rarefaction wave and one-shock wave tend to the vacuum solution and the contact discontinuity solution to the zero pressure relativistic Euler system, respectively.

scholarly journals Interactions of Delta Shock Waves for Zero-Pressure Gas Dynamics with Energy Conservation Law

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Wei Cai ◽  
Yanyan Zhang
Keyword(s):  
Theoretical Analysis ◽  
Shock Waves ◽  
Initial Data ◽  
Gas Dynamics ◽  
Riemann Solutions ◽  
Piecewise Constant ◽  
Delta Shock ◽  
Vacuum States ◽  
Delta Shock Waves ◽  
Energy Conservation Law

We study the interactions of delta shock waves and vacuum states for the system of conservation laws of mass, momentum, and energy in zero-pressure gas dynamics. The Riemann problems with initial data of three piecewise constant states are solved case by case, and four different configurations of Riemann solutions are constructed. Furthermore, the numerical simulations completely coinciding with theoretical analysis are shown.

scholarly journals On a Nonsymmetric Keyfitz-Kranzer System of Conservation Laws with Generalized and Modified Chaplygin Gas Pressure Law

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Hongjun Cheng ◽  
Hanchun Yang
Keyword(s):  
Shock Wave ◽  
Conservation Laws ◽  
Chaplygin Gas ◽  
Gas Pressure ◽  
Modified Chaplygin Gas ◽  
Delta Shock Wave ◽  
Generalized Chaplygin Gas ◽  
Riemann Solutions ◽  
Delta Shock ◽  
System Of Conservation Laws

This paper is devoted to the study of a nonsymmetric Keyfitz-Kranzer system of conservation laws with the generalized and modified Chaplygin gas pressure law, which may admit delta shock waves, a topic of interest. Firstly, we solve the Riemann problems with piecewise constant data having a single discontinuity. For the generalized Chaplygin gas pressure law, the solution consists of three different structures:R+J,S+J, andδ. Existence and uniqueness of delta shock solution are established under the generalized Rankine-Hugoniot relation and entropy condition. For the modified Chaplygin gas pressure law, the structures of solution areR+JandS+J. Secondly, we discuss the limits of Riemann solutions for the modified Chaplygin gas pressure law as the pressure law tends to the generalized Chaplygin gas one. In particular, for some cases, the solutionS+Jtends to a delta shock wave, and it is different from the delta shock wave for the generalized Chaplygin gas pressure law with the same initial data. Thirdly, we simulate the Riemann solutions and examine the formation process of delta shock wave by employing the Nessyahu-Tadmor scheme. The numerical results are coincident with the theoretical analysis.

scholarly journals On the delta shockwave interactions for the isentropic Chaplygin gas system consisting of three scalar equations

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5355-5373 ◽  
Author(s):  
Meina Sun ◽  
Jie Xin
Keyword(s):  
Shock Wave ◽  
Initial Data ◽  
Riemann Problem ◽  
Propagation Speed ◽  
Chaplygin Gas ◽  
Euler System ◽  
Delta Shock Wave ◽  
Riemann Solutions ◽  
Delta Shock ◽  
Scalar Equations

The Riemann problem for the one-dimensional version of isentropic compressible Euler system for the Chaplygin gas consisting of three scalar equations is considered. It is shown that the Riemann solutions involve only two situations: the combination of three contact discontinuities or a delta shock wave. The generalized Rankine-Hugoniot conditions of delta shock wave are derived and the exact delta shock wave solution including the strength and propagation speed is obtained explicitly. The solutions to the perturbed Riemann problem are constructed globally when the initial data are taken to be the three piecewise constant initial data. The wave interaction problem is extensively investigated and some interesting phenomena are observed. It is shown that the limits of solutions to the perturbed Riemann problem converge to the corresponding ones to the Riemann problem when the perturbation parameter tends to zero.

scholarly journals The Vanishing Pressure Limit of Riemann Solutions to the Non-Isentropic Euler Equations for Generalized Chaplygin Gas

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Qixia Ding ◽  
Lihui Guo
Keyword(s):  
Shock Wave ◽  
Contact Discontinuity ◽  
Vacuum State ◽  
Chaplygin Gas ◽  
Delta Shock Wave ◽  
Generalized Chaplygin Gas ◽  
Riemann Solutions ◽  
Pressure Limit ◽  
Delta Shock ◽  
Riemann Solution

We analyze the appearance of delta shock wave and vacuum state in the vanishing pressure limit of Riemann solutions to the non-isentropic generalized Chaplygin gas equations. As the pressure vanishes, the Riemann solution including two shock waves and possible one contact discontinuity converges to a delta shock wave solution. Both the densityρand the internal energyHsimultaneously present a Dirac delta singularity. And the Riemann solution involving two rarefaction waves and possible one contact discontinuity converges to a solution involving vacuum state of the transport equations.

Sign in / Sign up

Export Citation Format

Share Document