Concentration in the zero-exponent limit of solutions to the isentropic Euler equations for extended Chaplygin gas

2020 ◽  
pp. 1-33
Author(s):  
Yu Zhang ◽  
Yanyan Zhang ◽  
Jinhuan Wang
Keyword(s):  
Euler Equations ◽  
Chaplygin Gas ◽  
Isentropic Euler Equations

scholarly journals Consistent segregated staggered schemes with explicit steps for the isentropic and full Euler equations

2018 ◽  
Vol 52 (3) ◽  
pp. 893-944 ◽  
Author(s):  
Raphaèle Herbin ◽  
Jean-Claude Latché ◽  
Trung Tan Nguyen
Keyword(s):  
Energy Balance ◽  
Kinetic Energy ◽  
Internal Energy ◽  
Euler Equations ◽  
Left Hand ◽  
Isentropic Euler Equations ◽  
Space Discretization ◽  
Face Centered ◽  
The Stability ◽  
Stability And Consistency

In this paper, we build and analyze the stability and consistency of decoupled schemes, involving only explicit steps, for the isentropic Euler equations and for the full Euler equations. These schemes are based on staggered space discretizations, with an upwinding performed with respect to the material velocity only. The pressure gradient is defined as the transpose of the natural velocity divergence, and is thus centered. The velocity convection term is built in such a way that the solutions satisfy a discrete kinetic energy balance, with a remainder term at the left-hand side which is shown to be non-negative under a CFL condition. In the case of the full Euler equations, we solve the internal energy balance, to avoid the space discretization of the total energy, whose expression involves cell-centered and face-centered variables. However, since the residual terms in the kinetic energy balance (probably) do not tend to zero with the time and space steps when computing shock solutions, we compensate them by corrective terms in the internal energy equation, to make the scheme consistent with the conservative form of the continuous problem. We then show, in one space dimension, that, if the scheme converges, the limit is indeed an entropy weak solution of the system. In any case, the discretization preserves by construction the convex of admissible states (positivity of the density and, for Euler equations, of the internal energy), under a CFL condition. Finally, we present numerical results which confort this theory.

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.

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