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François Bouchut; Yann Jobic; Roberto Natalini; René Occelli; Vincent Pavan
Second-order entropy satisfying BGK-FVS schemes for incompressible Navier-Stokes equations
SMAI-Journal of computational mathematics, 4 (2018), p. 1-56
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Class. Math.: 65M12, 65X68
Keywords: incompressible Navier-Stokes equations, vector BGK schemes, flux vector splitting, low Mach number limit, discrete entropy inequality, cell Reynolds number, lattice Boltzmann schemes


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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