Asymptotic-preserving schemes for kinetic-fluid modeling of disperse two-phase flows

Goudon, Thierry; Jin, Shi; Liu, Jian Guo; Yan, Bokai

doi:10.1016/j.jcp.2013.03.038

Cited by 15 publications

(42 citation statements)

References 46 publications

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“…This choice of splitting parameter is illustrated by the idea in [19]. Similar to the first order scheme, this choice of α gives a stronger AP property as in (45).…”

Section: A Second Order Successive Penalty Methodsmentioning

confidence: 99%

A Successive Penalty-Based Asymptotic-Preserving Scheme for Kinetic Equations

Yan¹,

Jin²

2013

SIAM J. Sci. Comput.

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We propose an asymptotic-preserving (AP) scheme for kinetic equations that is efficient also in the hydrodynamic regimes. This scheme is based on the BGK-penalty method introduced by , but uses the penalization successively to achieve the desired asymptotic property. This method possesses a stronger AP property than the original method of Filbet-Jin, with the additional feature of being also positivity preserving when applied on the Boltzmann equation. It is also general enough to be applicable to several important classes of kinetic equations, including the Boltzmann equation and the Landau equation. Numerical experiments verify these properties.

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“…This choice of splitting parameter is illustrated by the idea in [19]. Similar to the first order scheme, this choice of α gives a stronger AP property as in (45).…”

Section: A Second Order Successive Penalty Methodsmentioning

confidence: 99%

A Successive Penalty-Based Asymptotic-Preserving Scheme for Kinetic Equations

Yan¹,

Jin²

2013

SIAM J. Sci. Comput.

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“…Another way to extend to second order with AP property is to use back-differentiation method (BDF) for the convection term, as was suggested in [8].…”

Section: Remark 42mentioning

confidence: 99%

A numerical scheme for the quantum Boltzmann equation with stiff collision terms

Filbet

Jin

2011

ESAIM: M2AN

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Abstract.Numerically solving the Boltzmann kinetic equations with the small Knudsen number is challenging due to the stiff nonlinear collision terms. A class of asymptotic-preserving schemes was introduced in [F. Filbet and S. Jin, J. Comput. Phys. 229 (2010) 7625-7648] to handle this kind of problems. The idea is to penalize the stiff collision term by a BGK type operator. This method, however, encounters its own difficulty when applied to the quantum Boltzmann equation. To define the quantum Maxwellian (Bose-Einstein or Fermi-Dirac distribution) at each time step and every mesh point, one has to invert a nonlinear equation that connects the macroscopic quantity fugacity with density and internal energy. Setting a good initial guess for the iterative method is troublesome in most cases because of the complexity of the quantum functions (Bose-Einstein or Fermi-Dirac function). In this paper, we propose to penalize the quantum collision term by a 'classical' BGK operator instead of the quantum one. This is based on the observation that the classical Maxwellian, with the temperature replaced by the internal energy, has the same first five moments as the quantum Maxwellian. The scheme so designed avoids the aforementioned difficulty, and one can show that the density distribution is still driven toward the quantum equilibrium. Numerical results are presented to illustrate the efficiency of the new scheme in both the hydrodynamic and kinetic regimes. We also develop a spectral method for the quantum collision operator.

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“…The density ρ ( t , x ) and velocity u ( t , x ) of the fluid are governed by the INS system

({, \begin{matrix} falsenonefalsearray \\ arraycenter \partial_{t} ρ + \nabla_{x} \cdot MathClass-open(ρu MathClass-close) = 0 \\ arraycenter \partial_{t} MathClass-open(ρu MathClass-close) + \nabla_{x} \cdot MathClass-open(ρu \otimes u MathClass-close) + \nabla_{x} p - \frac{1}{Re} Δ_{x} u + ρ \nabla_{x} Ψ = \frac{κ}{ϵ} \int MathClass-open(v - u MathClass-close) f d v, \\ arraycenter \nabla_{x} \cdot u = 0 . \end{matrix})

The pressure p ( t , x ) is determined by the divergence free condition. The readers are referred to and the references therein for the projection methods on variable density INS system.…”

Section: Introductionmentioning

confidence: 99%

“…Consequently, assuming that the dense phase is initially homogeneous, it remains homogeneous forever: if

ρ |_{t = 0} = \bar{ρ} > 0

is constant, then

ρ (t MathClass-punc, x) MathClass-rel= \overset{ρ}{true}

. This specific situation is investigated in the companion paper . However, the restriction to homogeneous fluid flows is unrealistic for most of the applications of practical interest.…”

Section: Introductionmentioning

confidence: 99%

“…We refer to for a recent review on the AP schemes and their applications. Different from , here, we need to develop the scheme in the framework of projection method for variable density INS, which requires different spatial discretization. The overall cost is comparable to solving one decoupled Vlasov–Fokker–Planck equation, see , and an INS by the projection method independently.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic‐preserving schemes for kinetic–fluid modeling of disperse two‐phase flows with variable fluid density

Goudon

Jin

Liu

et al. 2014

Numerical Methods in Fluids

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SUMMARYWe are concerned with a coupled system describing the interaction between suspended particles and a dense fluid. The particles are modeled by a kinetic equation of Vlasov–Fokker–Planck type, and the fluid is described by the incompressible Navier–Stokes system, with variable density. The systems are coupled through drag forces. High friction regimes lead to a purely hydrodynamic description of the mixture. We design first and second order asymptotic‐preserving schemes suited to such regimes. We extend the method introduced in [Goudon T, Jin S, Liu JG, Yan B. Journal of Computational Physics 2013; 246:145‐164] to the case of variable density in compressible flow. We check the accuracy and the asymptotic‐preserving property numerically. We set up a few numerical experiments to demonstrate the ability of the scheme in capturing intricate interactions between the two phases on a wide range of physical parameters and geometric situations. Copyright © 2014 John Wiley & Sons, Ltd.

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Asymptotic-preserving schemes for kinetic-fluid modeling of disperse two-phase flows

Cited by 15 publications

References 46 publications

A Successive Penalty-Based Asymptotic-Preserving Scheme for Kinetic Equations

A Successive Penalty-Based Asymptotic-Preserving Scheme for Kinetic Equations

A numerical scheme for the quantum Boltzmann equation with stiff collision terms

Asymptotic‐preserving schemes for kinetic–fluid modeling of disperse two‐phase flows with variable fluid density

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