scholarly journals Entropy stable, robust and high-order DGSEM for the compressible multicomponent Euler equations

2021 ◽  
pp. 110584
Author(s):  
Florent Renac
Keyword(s):  
Euler Equations ◽  
High Order ◽  
Entropy Stable

scholarly journals Entropy stable numerical approximations for the isothermal and polytropic Euler equations

2019 ◽  
Vol 60 (3) ◽  
pp. 791-824
Author(s):  
Andrew R. Winters ◽  
Christof Czernik ◽  
Moritz B. Schily ◽  
Gregor J. Gassner
Keyword(s):  
Total Energy ◽  
Euler Equations ◽  
Mass Conservation ◽  
Spectral Element ◽  
Momentum Conservation ◽  
High Order ◽  
Polytropic Model ◽  
Energy Behavior ◽  
Polytropic Equation ◽  
Entropy Stable

AbstractIn this work we analyze the entropic properties of the Euler equations when the system is closed with the assumption of a polytropic gas. In this case, the pressure solely depends upon the density of the fluid and the energy equation is not necessary anymore as the mass conservation and momentum conservation then form a closed system. Further, the total energy acts as a convex mathematical entropy function for the polytropic Euler equations. The polytropic equation of state gives the pressure as a scaled power law of the density in terms of the adiabatic index $$\gamma $$ γ . As such, there are important limiting cases contained within the polytropic model like the isothermal Euler equations ($$\gamma {=}1$$ γ = 1 ) and the shallow water equations ($$\gamma {=}2$$ γ = 2 ). We first mimic the continuous entropy analysis on the discrete level in a finite volume context to get special numerical flux functions. Next, these numerical fluxes are incorporated into a particular discontinuous Galerkin (DG) spectral element framework where derivatives are approximated with summation-by-parts operators. This guarantees a high-order accurate DG numerical approximation to the polytropic Euler equations that is also consistent to its auxiliary total energy behavior. Numerical examples are provided to verify the theoretical derivations, i.e., the entropic properties of the high order DG scheme.

scholarly journals Compact high-order schemes for the Euler equations

1988 ◽  
Vol 3 (3) ◽  
pp. 275-288 ◽  
Author(s):  
Saul Abarbanel ◽  
Ajay Kumar
Keyword(s):  
Euler Equations ◽  
High Order ◽  
High Order Schemes

scholarly journals Entropy stable essentially nonoscillatory methods based on RBF reconstruction

2019 ◽  
Vol 53 (3) ◽  
pp. 925-958 ◽  
Author(s):  
Jan S. Hesthaven ◽  
Fabian Mönkeberg
Keyword(s):  
Radial Basis Functions ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Basis Functions ◽  
One Dimensional ◽  
Hyperbolic Conservation ◽  
Radial Basis ◽  
Sign Property ◽  
Entropy Stable ◽  
Essentially Nonoscillatory

To solve hyperbolic conservation laws we propose to use high-order essentially nonoscillatory methods based on radial basis functions. We introduce an entropy stable arbitrary high-order finite difference method (RBF-TeCNOp) and an entropy stable second order finite volume method (RBF-EFV2) for one-dimensional problems. Thus, we show that methods based on radial basis functions are as powerful as methods based on polynomial reconstruction. The main contribution is the construction of an algorithm and a smoothness indicator that ensures an interpolation function which fulfills the sign-property on general one dimensional grids.

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