Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution

McDonald, James G.; Groth, C. P. T.

doi:10.1007/s00161-012-0252-y

Cited by 49 publications

(44 citation statements)

References 40 publications

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“…The distribution function derived, verifies a minimum entropy property and the consistency with the set of moments. Fundamental mathematical properties [42,66] such as positivity of the distribution function, hyperbolicity and entropy dissipation can be exhibited. Levermore [61] proposed a hierarchy of minimum-entropy closure where the lowest order closure are the Maxwellian and Gaussian closure.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic-preserving well-balanced scheme for the electronic M₁ model in the diffusive limit: Particular cases

et al. 2017

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Abstract. This work is devoted to the derivation of an asymptotic-preserving scheme for the electronic M1 model in the diffusive regime. The case without electric field and the homogeneous case are studied. The derivation of the scheme is based on an approximate Riemann solver where the intermediate states are chosen consistent with the integral form of the approximate Riemann solver. This choice can be modified to enable the derivation of a numerical scheme which also satisfies the admissible conditions and is well-suited for capturing steady states. Moreover, it enjoys asymptotic-preserving properties and handles the diffusive limit recovering the correct diffusion equation. Numerical tests cases are presented, in each case, the asymptotic-preserving scheme is compared to the classical HLL [A. Harten, P.D. Lax and B. Van Leer, SIAM Rev. 25 (1983) 35-61.] scheme usually used for the electronic M1 model. It is shown that the new scheme gives comparable results with respect to the HLL scheme in the classical regime. On the contrary, in the diffusive regime, the asymptotic-preserving scheme coincides with the expected diffusion equation, while the HLL scheme suffers from a severe lack of accuracy because of its unphysical numerical viscosity.Mathematics Subject Classification. 65C20, 65M12.

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Section: Introductionmentioning

confidence: 99%

Asymptotic-preserving well-balanced scheme for the electronic M₁ model in the diffusive limit: Particular cases

et al. 2017

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“…In contrast to the closure proposed in [6], the β-closure can be used directly in any standard numerical scheme for hyperbolic systems such as the finite volume method. A publication with a detailed study of the β-closure is in preparation.…”

Section: Moment Equations For Non-equilibrium Gas Flowsmentioning

confidence: 99%

“…Recently, a closed-form closure for the 5-moment equations, featuring some characteristics of the corresponding maximumentropy closure, was put forward by M c Donald in [6] with very promising results for hypersonic shock-wave simulations. Just as the maximum-entropy closure, the closed-form closure shares the curious feature, thatŜ is singular on a half-line L emanating from the equlibrium state; cf.…”

Section: Moment Equations For Non-equilibrium Gas Flowsmentioning

confidence: 99%

“…[6,7]. In the special case β = 0, the closure reduces to a form similar to the closure proposed in [6] and S 0 is singular on the half-line L. Hyperbolicity of this closure has been verified numerically on the subset {(1, 0, 1,Q,R) : 1 +Q 2 <R ≤ 10 3 } with 0 ≤ β 0.7 of normalized moments, allowing the simulation of flows with strong deviations from local thermodynamic equilibrium.…”

Section: Moment Equations For Non-equilibrium Gas Flowsmentioning

confidence: 99%

“…While these closures yield globally hyperbolic systems, the evaluation of the closing fluxes requires the use of computationally expensive optimization methods if the equations include terms higher than the second moment [4][5][6].…”

Section: Moment Equations For Non-equilibrium Gas Flowsmentioning

confidence: 99%

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A new closure mitigating the sub‐shock phenomenon in the continuous shock‐structure problem

Schärer

Torrilhon

2013

Proc Appl Math and Mech

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Moment equations offer an attractive framework for modeling non-equilibrium processes in moderately rarefied gases. We present a new hyperbolic closure for the 5-moment system in the one-dimensional setting that couples the conservation laws of mass, momentum and energy to two evolution equations of higher, non-equilibrium moments. The new closure, given in closed-form, depends on a regularization parameter β that is related to the non-linearity around the equilibrium state, so as to mitigate the appearing sub-shock in the continuous shock-structure problem.We consider the one-particle phase space f :where c denotes the microscopic velocity, such that n(x, t) = f := R f (x, t, c) dc is the particle density. Moments of the distribution function f over velocity space yield macroscopic quantities such as the mass density ρ = m f , macroscopic velocity v = m cf /ρ or temperature θ = m C 2 f /ρ in energy units, where C = c − v denotes the random velocity and m is the particle mass. We use the simplified BGK operator [1] to model the effect of particle collisions. The evolution of f is then described by the Boltzmann BGK equation(1) denotes the local Maxwell-Boltzmann equilibrium distribution function and τ is the relaxation time. We further define the higher non-convective moments Q = m C 3 f , R = m C 4 f , S = m C 5 f , such that q = Q/2 is the physical heat flux. For any fixed time t and position x, the distribution function, f , can be normalized by the mapping f (c) = (n/θ, such that the normalized momentsû = ξ if with i = 0, . . . , 4 are given byû = (1, 0, 1,Q,R) T , whereQ = Q/(ρθ 3/2 ),R = R/(ρθ 2 ). Let u = (ρ, v, θ,Q,R) T denote the vector of field variables, then we can express the convective moments as E i (u) = c i f for i = 0, . . . , 4. The projection of equation (1) on these convective moments yields the 5-moment system in balance law formIn order to close the 5-moment system a constitutive relation between the non-convective momentŜ = S/(ρθ 5/2 ) and the field variables u is required. Such relations can be found by the postulation of a model distribution function,f (Model) , parameterized so that the moment constraints ξ if (Model) =û i for i = 0, . . . , 4 uniquely determinef (Model) and thus alsô S = ξ 5f (Model) . The now classical closure proposed by Grad in [2] is based on a Hermite expansion of the distribution function in velocity space, which yields the simple closureŜ (Grad) = 10Q for the 5-moment system. Unfortunately, the resulting system is only hyperbolic for small deviations from the local equilibrium distribution, greatly limiting the range of processes that can be modeled with Grad's closure.Closures based on the maximization of the physical entropy have been proposed [5,8]. While these closures yield globally hyperbolic systems, the evaluation of the closing fluxes requires the use of computationally expensive optimization methods if the equations include terms higher than the second moment [4][5][6].Recently, a closed-form closure for the 5-moment equations, featuring some character...

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RET of Rarefied Monatomic Gas: Non-relativistic Theory

Ruggeri

Sugiyama

2020

Classical and Relativistic Rational Extended Thermodynamics of Gases

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Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution

Cited by 49 publications

References 40 publications

Asymptotic-preserving well-balanced scheme for the electronic M₁ model in the diffusive limit: Particular cases

Asymptotic-preserving well-balanced scheme for the electronic M₁ model in the diffusive limit: Particular cases

A new closure mitigating the sub‐shock phenomenon in the continuous shock‐structure problem

RET of Rarefied Monatomic Gas: Non-relativistic Theory

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Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution

Cited by 49 publications

References 40 publications

Asymptotic-preserving well-balanced scheme for the electronic M1 model in the diffusive limit: Particular cases

Asymptotic-preserving well-balanced scheme for the electronic M1 model in the diffusive limit: Particular cases

A new closure mitigating the sub‐shock phenomenon in the continuous shock‐structure problem

RET of Rarefied Monatomic Gas: Non-relativistic Theory

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Product

Resources

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Asymptotic-preserving well-balanced scheme for the electronic M₁ model in the diffusive limit: Particular cases

Asymptotic-preserving well-balanced scheme for the electronic M₁ model in the diffusive limit: Particular cases