A discrete kinetic approximation for the incompressible Navier-Stokes equations

Carfora, Maria Francesca; Natalini, Roberto

doi:10.1051/m2an:2007055

Cited by 14 publications

(16 citation statements)

References 25 publications

(17 reference statements)

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“…However, discrete effects in the singular hydrodynamic limit lead to the important fact that the consistency, which is related to moment relations, has to be examined at the discrete level and cannot be deduced from the continuous approach. This is what happens in [16, section 6] and in [4,19,29]. In this context not only the Laplacian but also cross derivatives can be handled, see [19].…”

Section: Parabolic Scalingmentioning

confidence: 69%

“…By analyzing the simultaneous hydrodynamic and low Mach number limits, we establish the properties for having consistency, and second-order accuracy. Our work takes its roots in [29] where finitely many vector Maxwellians are used, in accordance with [16]. Even if we start from kinetic considerations, we write the scheme under the form of a simple explicit finite volume/difference flux vector splitting scheme over a Cartesian mesh and written on the macroscopic moments themselves, thus finally avoiding the kinetic aspects.…”

Section: The Kinetic Bgk Methodsmentioning

confidence: 99%

“…In the time-space discrete case where the collision term is replaced by the projection onto Maxwellians at discrete times, the second-order viscous terms in the limit equations on the moments derive indeed from the numerical viscosity of the hydrodynamic scheme, see [16,Section 6], [4,19,29].…”

Section: Flux Vector Splitting Schemes For Incompressible Navier-stokmentioning

confidence: 99%

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Second-order entropy satisfying BGK-FVS schemes for incompressible Navier-Stokes equations

Bouchut¹,

Jobic²,

Natalini³

et al. 2018

The SMAI Journal of Computational Mathematics

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Abstract. Kinetic BGK numerical schemes for the approximation of incompressible Navier-Stokes equations are derived via classical discrete velocity vector BGK approximations, but applied to an inviscid compressible gas dynamics system with small Mach number parameter, according to the approach of Carfora and Natalini (2008). As the Mach number, the grid size and the timestep tend to zero, the low Mach number limit and the time-space convergence of the scheme are achieved simultaneously, and the numerical viscosity tends to the physical viscosity of the Navier-Stokes system. The method is analyzed and formulated as an explicit finite volume/difference flux vector splitting (FVS) scheme over a Cartesian mesh. It is close in spirit to lattice Boltzmann schemes, but it has several advantages. The first is that the scheme is expressed only in terms of momentum and mass compressible variables. It is therefore very easy to implement, and several types of boundary conditions are straightforward to apply. The second advantage is that the scheme satisfies a discrete entropy inequality, under a CFL condition of parabolic type and a subcharacteristic stability condition involving a cell Reynolds number that ensures that diffusion dominates advection at the level of the grid size. This ensures the robustness of the method, with explicit uniform bounds on the approximate solution. Moreover the scheme is proved to be second-order accurate in space if the parameters are well chosen, this is the case in particular for the Lax-Friedrichs scheme with Mach number proportional to the grid size. The scheme falls then into the class of artificial compressibility methods, the novelty being its exceptionally good theoretical properties. We show the efficiency of the method in terms of accuracy and robustness on a variety of classical two-dimensional benchmark tests. The method is finally applied in three dimensions to compute the permeability of a porous medium defined by a complex idealized Kelvin-like cell. Relations between our scheme and compressible low Mach number schemes are discussed.

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Section: Parabolic Scalingmentioning

confidence: 69%

Section: The Kinetic Bgk Methodsmentioning

confidence: 99%

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Second-order entropy satisfying BGK-FVS schemes for incompressible Navier-Stokes equations

Bouchut¹,

Jobic²,

Natalini³

et al. 2018

The SMAI Journal of Computational Mathematics

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“…and u ε = L i=1 f i is the approximating vector field, converging to the solution to the limit system, which is established under some consistency conditions, see [8,13,1,2,10] for a detailed discussion. An important feature of these approximations is the existence, under some reasonable conditions, of a kinetic entropy.…”

Section: Kinetic Entropies and The Relative Entropymentioning

confidence: 99%

Strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations via the relative entropy method

Bianchini

2019

Journal de Mathématiques Pures et Appliquées

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The aim of this paper is to prove the strong convergence of the solutions to a vector-BGK model under the diffusive scaling to the incompressible Navier-Stokes equations on the two-dimensional torus. This result holds in any interval of time [0, T ], with T > 0. We also provide the global in time uniform boundedness of the solutions to the approximating system. Our argument is based on the use of local in time H s -estimates for the model, established in a previous work, combined with the L 2 -relative entropy estimate and the interpolation properties of the Sobolev spaces.

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“…In [13,10], it is numerically studied the convergence of the solutions to the vector BGK model to the solutions to the incompressible Navier-Stokes equations. More precisely, assuming that, in a suitable functional space, ρ ε →ρ, u ε →û, and ρ ε −ρ ε 2 →P , under some consistency conditions of the BGK approximation with respect to the Navier-Stokes equations, [13], it can be shown that the couple (û,P ) is a solution to the incompressible Navier-Stokes equations. The aim of the present paper is to provide a rigorous proof of this convergence in the Sobolev spaces.…”

Section: Introductionmentioning

confidence: 99%

Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations

Bianchini¹,

Natalini²

2019

Kinetic &Amp; Related Models

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We present a rigorous convergence result for the smooth solutions to a singular semilinear hyperbolic approximation, a vector BGK model, to the solutions to the incompressible Navier-Stokes equations in Sobolev spaces. Our proof is based on the use of a constant right symmetrizer, weighted with respect to the parameter of the singular pertubation system. This symmetrizer provides a conservative-dissipative form for the system and this allow us to perform uniform energy estimates and to get the convergence by compactness.

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A discrete kinetic approximation for the incompressible Navier-Stokes equations

Cited by 14 publications

References 25 publications

Second-order entropy satisfying BGK-FVS schemes for incompressible Navier-Stokes equations

Second-order entropy satisfying BGK-FVS schemes for incompressible Navier-Stokes equations

Strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations via the relative entropy method

Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations

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