A minimum entropy principle in the compressible multicomponent Euler equations

Gouasmi, Ayoub; Duraisamy, Karthik; Murman, Scott M.; Tadmor, Eitan

doi:10.1051/m2an/2019070

Cited by 12 publications

(15 citation statements)

References 29 publications

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“…As observed in [31], the function H 0 (S), although not smooth, can be written as the limit of a sequence of smooth functions satisfying (14); see also [6,Section 3.1] for a detailed review. Therefore, by passing to the limit, the inequality (26) holds for H = H 0 , which gives…”

Section: Minimum Principle Of the Specific Entropymentioning

confidence: 99%

“…Jiang and Liu proposed new IRP limiters for the DG schemes to the isentropic Euler equations [20], the compressible Euler equations [17], and general multi-dimensional hyperbolic conservation laws [18]. Gouasmi et al [6] proved a minimum entropy principle on entropy solutions to the compressible multicomponent Euler equations at the smooth and discrete levels.…”

Section: Introductionmentioning

confidence: 99%

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Minimum Principle on Specific Entropy and High-Order Accurate Invariant Region Preserving Numerical Methods for Relativistic Hydrodynamics

Wu¹

2021

Preprint

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This paper first explores Tadmor's minimum entropy principle for the special relativistic hydrodynamics (RHD) equations and incorporates this principle into the design of robust highorder discontinuous Galerkin (DG) and finite volume schemes for RHD on general meshes. The proposed schemes are rigorously proven to preserve numerical solutions in a global invariant region constituted by all the known intrinsic constraints: minimum entropy principle, the subluminal constraint on fluid velocity, the positivity of pressure, and the positivity of restmass density. Relativistic effects lead to some essential difficulties in the present study, which are not encountered in the non-relativistic case. Most notably, in the RHD case the specific entropy is a highly nonlinear implicit function of the conservative variables, and, moreover, there is also no explicit formula of the flux in terms of the conservative variables. In order to overcome the resulting challenges, we first propose a novel equivalent form of the invariant region, by skillfully introducing two auxiliary variables. As a notable feature, all the constraints in the novel form are explicit and linear with respect to the conservative variables. This provides a highly effective approach to theoretically analyze the invariant-region-preserving (IRP) property of numerical schemes for RHD, without any assumption on the IRP property of the exact Riemann solver. Based on this, we prove the convexity of the invariant region and establish the generalized Lax-Friedrichs splitting properties via technical estimates, lying the foundation for our rigorous IRP analyses. It is rigorously shown that the first-order Lax-Friedrichs type scheme for the RHD equations satisfies a local minimum entropy principle and is IRP under a CFL condition. Provably IRP high-order accurate DG and finite volume methods are then developed for the RHD with the help of a simple scaling limiter, which is designed by following the bound-preserving type limiters in the literature. Several numerical examples demonstrate the effectiveness and robustness of the proposed schemes.

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Section: Minimum Principle Of the Specific Entropymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Minimum Principle on Specific Entropy and High-Order Accurate Invariant Region Preserving Numerical Methods for Relativistic Hydrodynamics

Wu¹

2021

Preprint

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“…Numerical experiments showed that the ES scheme we constructed is no exception. We stress that these anomalies, which are not present in the single component case, violate neither entropy stability nor a minimum principle of the specific entropy [64]. The remedies to the pressure oscillations problem typically consist in giving up on conservation of total energy [45,44,38,43], which can impair the ability of the scheme to properly capture shocks.…”

Section: Discussionmentioning

confidence: 96%

Formulation of Entropy-Stable schemes for the multicomponent compressible Euler equations

Gouasmi¹,

Duraisamy²,

Murman³

2020

Computer Methods in Applied Mechanics and Engineering

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In this work, Entropy-Stable (ES) schemes are formulated for the multicomponent compressible Euler equations. Entropy-conservative (EC) and ES fluxes are derived. Particular attention is paid to the limit case of zero partial densities where the structure required by ES schemes breaks down (the entropy variables are no longer defined). It is shown that while an EC flux is well-defined in this limit, a well-defined upwind ES flux requires appropriately averaged partial densities in the dissipation matrix. A similar result holds for the high-order TecNO reconstruction. However, this does not prevent the numerical solution from developing negative partial densities or internal energy. Numerical experiments were performed on one-dimensional and two-dimensional interface and shock-interface problems. The present scheme exactly preserves stationary interfaces. On moving interfaces, it produces spurious pressure oscillations typically observed with conservative schemes [Karni, J. Comput. Phys., 112 (1994) 1]. We find that these anomalies, which are not present in the single-component case, violate neither entropy stability nor a minimum principle of the specific entropy. Finally, we show that the scheme is able to reproduce the physical mechanisms of the two-dimensional shock-bubble interaction problem [Haas & Sturtevant,

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“…Proof Twice differentiability is straightforward from (2.16). To prove the convexity we use the trick introduced in [28] and also used in [24] to prove that the Hessian of the entropy H η is congruent to the following strictly convex diagonal matrix:…”

Section: Entropy Pairmentioning

confidence: 99%

Energy relaxation approximation for the compressible multicomponent flows in thermal nonequilibrium

Marmignon,

Naddei,

Renac

2021

Preprint

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This work concerns the numerical approximation with a finite volume method of inviscid, nonequilibrium, high-temperature flows in multiple space dimensions. It is devoted to the analysis of the numerical scheme for the approximation of the hyperbolic system in homogeneous form. We derive a general framework for the design of numerical schemes for this model from numerical schemes for the monocomponent compressible Euler equations for a polytropic gas. Under a very simple condition on the adiabatic exponent of the polytropic gas, the scheme for the multicomponent system enjoys the same properties as the one for the monocomponent system: discrete entropy inequality, positivity of the partial densities and internal energies, discrete maximum principle on the mass fractions, and discrete minimum principle on the entropy. Our approach extends the relaxation of energy [Coquel and Perthame, SIAM J. Numer. Anal., 35 (1998Anal., 35 ( ), 2223Anal., 35 ( -2249 to the multicomponent Euler system. In the limit of instantaneous relaxation we show that the solution formally converges to a unique and stable equilibrium solution to the multicomponent Euler equations. We then use this framework to design numerical schemes from three schemes for the polytropic Euler system: the Godunov exact Riemann solver [Godunov, Math. Sbornik, 47 (1959), 271-306] and the HLL [Harten et al., SIAM Rev., 25 (1983), 35-61] and pressure relaxation based [Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Frontiers in Mathematics, Birkhäuser, 2004] approximate Riemann solvers. Numerical experiments in one and two space dimensions on flows with discontinuous solutions support the conclusions of our analysis and highlight stability, robustness and convergence of the scheme.

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A minimum entropy principle in the compressible multicomponent Euler equations

Cited by 12 publications

References 29 publications

Minimum Principle on Specific Entropy and High-Order Accurate Invariant Region Preserving Numerical Methods for Relativistic Hydrodynamics

Minimum Principle on Specific Entropy and High-Order Accurate Invariant Region Preserving Numerical Methods for Relativistic Hydrodynamics

Formulation of Entropy-Stable schemes for the multicomponent compressible Euler equations

Energy relaxation approximation for the compressible multicomponent flows in thermal nonequilibrium

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