Wave Interactions and Stability of Riemann Solutions of the Aw–Rascle Model for Generalized Chaplygin Gas

2017 ◽  
Vol 154 (1) ◽  
pp. 95-109 ◽  
Author(s):  
Yujin Liu ◽  
Wenhua Sun
Keyword(s):  
Chaplygin Gas ◽  
Wave Interactions ◽  
Generalized Chaplygin Gas ◽  
Riemann Solutions

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.

scholarly journals Limit of Riemann Solutions to the Nonsymmetric System of Keyfitz-Kranzer Type

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Lihui Guo ◽  
Gan Yin
Keyword(s):  
Shock Wave ◽  
Euler Equations ◽  
Chaplygin Gas ◽  
Transport Equations ◽  
Delta Shock Wave ◽  
Generalized Chaplygin Gas ◽  
Riemann Solutions ◽  
Delta Shock ◽  
Nonsymmetric System ◽  
Isothermal Gas

The limit of Riemann solutions to the nonsymmetric system of Keyfitz-Kranzer type with a scaled pressure is considered for both polytropic gas and generalized Chaplygin gas. In the former case, the delta shock wave can be obtained as the limit of shock wave and contact discontinuity whenu->u+and the parameterϵtends to zero. The point is, the delta shock wave is not the one of transport equations, which is obviously different from cases of some other systems such as Euler equations or relativistic Euler equations. For the generalized Chaplygin gas, unlike the polytropic or isothermal gas, there exists a certain critical valueϵ2depending only on the Riemann initial data, such that whenϵdrops toϵ2, the delta shock wave appears asu->u+, which is actually a delta solution of the same system in one critical case. Then asϵbecomes smaller and goes to zero at last, the delta shock wave solution is the exact one of transport equations. Furthermore, the vacuum states and contact discontinuities can be obtained as the limit of Riemann solutions whenu-<u+andu-=u+, respectively.

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.

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