Delta-shocks and vacuums in the relativistic Euler equations for isothermal fluids with the flux approximation

2019 ◽  
Vol 60 (1) ◽  
pp. 011508 ◽  
Author(s):  
Yu Zhang ◽  
Yanyan Zhang
Keyword(s):  
Euler Equations ◽  
Relativistic Euler Equations ◽  
Flux Approximation ◽  
Delta Shocks

scholarly journals Delta Shocks and Vacuums to the Isentropic Euler Equations with the Flux Perturbation for van der Waals Gas

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Jinhuan Wang ◽  
Yongbin Nie ◽  
Samuele De Bartolo
Keyword(s):  
Euler Equations ◽  
Van Der Waals ◽  
Excluded Volume ◽  
Formation Mechanisms ◽  
Van Der Waals Gas ◽  
Riemann Solutions ◽  
Flux Approximation ◽  
Isentropic Euler Equations ◽  
Delta Shocks ◽  
Flux Perturbation

In this paper, we study the isentropic Euler equations with the flux perturbation for van der Waals gas, in which the density has both lower and upper bounds due to the introduction of the flux approximation and the molecular excluded volume. First, we solve the Riemann problem of this system and construct the Riemann solutions. Second, the formation mechanisms of delta shocks and vacuums are analyzed for the Riemann solutions as the pressure, the flux approximation, and the molecular excluded volume all vanish. Finally, some numerical simulations are demonstrated to verify the theoretical analysis.

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.

scholarly journals A new variation for the relativistic Euler equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mahmoud A. E. Abdelrahman ◽  
Hanan A. Alkhidhr
Keyword(s):  
Euler Equations ◽  
Nonlinear Waves ◽  
Approximation Scheme ◽  
Strength Function ◽  
Convergent Sequence ◽  
Approximate Solutions ◽  
Large Initial Data ◽  
Initial Value ◽  
Relativistic Euler Equations ◽  
Glimm Scheme

Abstract The Glimm scheme is one of the so famous techniques for getting solutions of the general initial value problem by building a convergent sequence of approximate solutions. The approximation scheme is based on the solution of the Riemann problem. In this paper, we use a new strength function in order to present a new kind of total variation of a solution. Based on this new variation, we use the Glimm scheme to prove the global existence of weak solutions for the nonlinear ultra-relativistic Euler equations for a class of large initial data that involve the interaction of nonlinear waves.

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