Delta Shock Wave Solution of the Riemann Problem for the Non-homogeneous Modified Chaplygin Gasdynamics

2020 ◽  
Author(s):  
Rahul Kumar Chaturvedi ◽  
L. P. Singh ◽  
Dia Zeidan
Keyword(s):  
Shock Wave ◽  
Riemann Problem ◽  
Wave Solution ◽  
Delta Shock Wave ◽  
Delta Shock ◽  
Shock Wave Solution

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 Regularization of the Shock Wave Solution to the Riemann Problem for the Relativistic Burgers Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Ting Zhang ◽  
Chun Shen
Keyword(s):  
Shock Wave ◽  
Initial Data ◽  
Riemann Problem ◽  
Wave Solution ◽  
Burgers Equation ◽  
Explicit Computation ◽  
Convective Term ◽  
Regularization Technique ◽  
Shock Wave Solution

The regularization of the shock wave solution to the Riemann problem for the relativistic Burgers equation is considered. For Riemann initial data consisting of a single decreasing jump, we find that the regularization of nonlinear convective term cannot capture the correct shock wave solution. In order to overcome it, we consider a new regularization technique called the observable divergence method introduced by Mohseni and discover that it can capture the correct shock wave solution. In addition, we take the Helmholtz filter for the fully explicit computation.

Riemann problem and limits of solutions to the isentropic relativistic Euler equations for isothermal gas with flux approximation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yu Zhang ◽  
Yanyan Zhang
Keyword(s):  
Shock Wave ◽  
Euler Equations ◽  
Riemann Problem ◽  
Delta Shock Wave ◽  
Relativistic Euler Equations ◽  
Delta Shock ◽  
Isentropic Relativistic Euler Equations ◽  
Flux Approximation ◽  
Two Parameter ◽  
Pressureless Relativistic Euler Equations

Abstract We are concerned with the vanishing flux-approximation limits of solutions to the isentropic relativistic Euler equations governing isothermal perfect fluid flows. The Riemann problem with a two-parameter flux approximation including pressure term is first solved. Then, we study the limits of solutions when the pressure and two-parameter flux approximation vanish, respectively. It is shown that, any two-shock-wave Riemann solution converges to a delta-shock solution of the pressureless relativistic Euler equations, and the intermediate density between these two shocks tends to a weighted δ-measure that forms a delta shock wave. By contract, any two-rarefaction-wave solution tends to a two-contact-discontinuity solution of the pressureless relativistic Euler equations, and the intermediate state in between tends to a vacuum state.

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.

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.

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