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.

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 Solutions with concentration and cavitation to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas

2019 ◽  
Vol 17 (1) ◽  
pp. 220-241 ◽  
Author(s):  
Yunfeng Zhang ◽  
Meina Sun ◽  
Xiuli Lin
Keyword(s):  
Riemann Problem ◽  
Vacuum State ◽  
Chaplygin Gas ◽  
Euler System ◽  
Phase Plane Analysis ◽  
Delta Shock Wave ◽  
Plane Analysis ◽  
Delta Shock ◽  
Riemann Solution ◽  
Shock Wave Solution

Abstract The solutions to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas are constructed for all kinds of situations by using the method of phase plane analysis. The asymptotic limits of solutions to the Riemann problem for the relativistic extended Chaplygin Euler system are investigated in detail when the pressure given by the equation of state of extended Chaplygin gas becomes that of the pressureless gas. During the process of vanishing pressure, the phenomenon of concentration can be identified and analyzed when the two-shock Riemann solution tends to a delta shock wave solution as well as the phenomenon of cavitation also being captured and observed when the two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution with a vacuum state between them.

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 limit Riemann solutions to nonisentropic Chaplygin Euler equations

2020 ◽  
Vol 18 (1) ◽  
pp. 1771-1787
Author(s):  
Maozhou Lin ◽  
Lihui Guo
Keyword(s):  
Shock Wave ◽  
Internal Energy ◽  
Euler Equations ◽  
Vacuum State ◽  
Entropy Inequality ◽  
Delta Shock Wave ◽  
Riemann Solutions ◽  
Delta Shock

Abstract We mainly consider the limit behaviors of the Riemann solutions to Chaplygin Euler equations for nonisentropic fluids. The formation of delta shock wave and the appearance of vacuum state are found as parameter ε \varepsilon tends to a certain value. Different from the isentropic fluids, the weight of delta shock wave is determined by variance density ρ \rho and internal energy H. Meanwhile, involving the entropy inequality, the uniqueness of delta shock wave is obtained.

Measure-Valued Solutions to a Non-Strictly Hyperbolic System with Delta-Type Riemann Initial Data

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Shuangrong Li ◽  
Chun Shen
Keyword(s):  
Shock Wave ◽  
Initial Data ◽  
Riemann Problem ◽  
Contact Discontinuity ◽  
Initial Condition ◽  
Global Measure ◽  
Delta Shock Wave ◽  
Dirac Delta ◽  
Delta Shock ◽  
Delta Type

AbstractThis paper is concerned with the construction of global measure-valued solutions to the extended Riemann problem for a non-strictly hyperbolic system of two conservation laws with delta-type initial data. The wave interaction problems have been extensively studied for all kinds of situations by using the initial condition consisting of constant states in three pieces instead of delta-type initial data under the perturbation method. The measure-valued solutions of the extended Riemann problem are achieved constructively when the perturbed parameter tends to zero. During the process of constructing solutions, a new and interesting nonlinear phenomenon is discovered, in which the initial Dirac delta function travels along the trajectory of either delta shock wave or contact discontinuity (or delta contact discontinuity). Moreover, a delta shock wave is separated into a delta contact discontinuity and a shock wave during the process of delta shock wave penetrating a composite wave composed of a rarefaction wave and a contact discontinuity. In addition, we further consider the constructions of global measure-valued solutions when the initial condition contains Dirac delta functions at two different initial points.

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