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

Abstract

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.

An entropy satisfying two-speed relaxation system for the barotropic Euler equations: application to the numerical approximation of low Mach number flows

“…When the Mach number is small a well-known difficulty is to design asymptoticpreserving schemes, which means schemes that are uniformly accurate with respect to the Mach number. It has been pointed out in [7] that until now there was no asymptotic-preserving scheme which satisfies a fully discrete entropy inequality and which is first-order accurate in the low Mach number regime. Indeed a scheme proposed in [7] is entropy-satisfying and asymptotic-preserving but the order of accuracy is only 2/3 in the incompressible asymptotics.…”

“…It has been pointed out in [7] that until now there was no asymptotic-preserving scheme which satisfies a fully discrete entropy inequality and which is first-order accurate in the low Mach number regime. Indeed a scheme proposed in [7] is entropy-satisfying and asymptotic-preserving but the order of accuracy is only 2/3 in the incompressible asymptotics. On the contrary, the Lagrange-Projection scheme proposed in [14] (see also [15]) is first-order accurate (uniformly with respect to the Mach number) but the underlying entropy inequality is no longer valid when we are too close to the asymptotic limit.…”

Determining permeability tensors of porous media: A novel ‘vector kinetic’ numerical approach

“…That means that the flux is decomposed in separated directions: ( ) = + ( ) + − ( ). We have chosen the Lax-Friedrichs decomposition in order to define Fj ± (W) (Bouchut, 2018). This will enable to import the continuous kinetic entropy condition to the discrete level.…”

“…As the scheme is only written in term of moments, the boundary conditions are treated as in FVM or finite difference method. For applying Dirichlet boundary conditions, we use the standard ghost cells Neumann to the other components (see Bouchut, 2018 for details).…”

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

“…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.…”