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 Concentration in vanishing adiabatic exponent limit of solutions to the Aw–Rascle traffic model

2021 ◽  
pp. 1-35
Author(s):  
Shouqiong Sheng ◽  
Zhiqiang Shao
Keyword(s):  
Shock Wave ◽  
Theoretical Analysis ◽  
Shock Waves ◽  
Gas Dynamics ◽  
Traffic Model ◽  
Adiabatic Exponent ◽  
Delta Shock Wave ◽  
Riemann Solutions ◽  
Distribution Sense ◽  
Delta Shock

In this paper, we study the phenomenon of concentration and the formation of delta shock wave in vanishing adiabatic exponent limit of Riemann solutions to the Aw–Rascle traffic model. It is proved that as the adiabatic exponent vanishes, the limit of solutions tends to a special delta-shock rather than the classical one to the zero pressure gas dynamics. In order to further study this problem, we consider a perturbed Aw–Rascle model and proceed to investigate the limits of solutions. We rigorously proved that, as the adiabatic exponent tends to one, any Riemann solution containing two shock waves tends to a delta-shock to the zero pressure gas dynamics in the distribution sense. Moreover, some representative numerical simulations are exhibited to confirm the theoretical analysis.

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 Delta Shock Waves for a Linearly Degenerate Hyperbolic System of Conservation Laws of Keyfitz-Kranzer Type

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Hongjun Cheng
Keyword(s):  
Shock Waves ◽  
Conservation Laws ◽  
Hyperbolic System ◽  
Delta Shock Wave ◽  
Riemann Solutions ◽  
Delta Shock ◽  
System Of Conservation Laws ◽  
Elementary Waves ◽  
Delta Shock Waves ◽  
Linearly Degenerate

This paper is devoted to the study of delta shock waves for a hyperbolic system of conservation laws of Keyfitz-Kranzer type with two linearly degenerate characteristics. The Riemann problem is solved constructively. The Riemann solutions include exactly two kinds. One consists of two (or just one) contact discontinuities, while the other contains a delta shock wave. Under suitable generalized Rankine-Hugoniot relation and entropy condition, the existence and uniqueness of delta shock solution are established. These analytical results match well the numerical ones. Finally, two kinds of interactions of elementary waves are discussed.

scholarly journals A Deposition Model: Riemann Problem and Flux-Function Limits of Solutions

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Hongjun Cheng ◽  
Shiwei Li
Keyword(s):  
Initial Data ◽  
Riemann Solutions ◽  
Limit System ◽  
Flux Function ◽  
Deposition Model ◽  
Delta Shock ◽  
Riemann Solution ◽  
Scalar Conservation Law ◽  
Vacuum States ◽  
Scalar Conservation

The Riemann solutions of a deposition model are shown. A singular flux-function limit of the obtained Riemann solutions is considered. As a result, it is shown that the Riemann solutions of the deposition model just converge to the Riemann solutions of the limit system, the scalar conservation law with a linear flux function involving discontinuous coefficient. Especially, for some initial data, the two-shock Riemann solution of the deposition model tends to the delta-shock Riemann solution of the limit system; by contrast, for some initial data, the two-rarefaction-wave Riemann solution of the deposition model tends to the vacuum Riemann solution of the limit system. Some numerical results exhibiting the formation processes of delta-shocks and vacuum states are presented.

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