During the past two decades, Lattice Boltzmann Methods (LBM) have been intensively developed and applied to various flow situations of industrial relevance such as in those in mixing, filtration and other chemical engineering operations [1, 2]. These methods have proven precise and robust. One major limitation, however, comes from the necessity to use regular grids. Therefore, complex geometries
involving, for instance, porous media are still challenging because lattice Boltzmann grids cannot be locally rened in a straightforward manner.
LBM methods can be seen as a particular discretization of kinetic equations, with a discrete set of given velocities and a collision term. The target equation (e.g. incompressible Navier-Stokes equations or any convection-diusion equation) is obtained by a limiting process involving dierent scales (diusive or fluid) and dimensionless numbers (Mach and Knudsen numbers, for example), [3].
It seems reasonable to expect from a numerical scheme, that it preserves the asymptotic behavior of the continuous equations in the following sense: the limiting process, when applied to the discrete equations, should lead to a numerical scheme that is consistent with the target equation. In this case, the numerical scheme is
referred to as asymptotic preserving, [4]. A wide class of asymptotic preserving schemes have been developed, but only a few have dealt with 2D or 3D unstructured grids [5, 6].
In this work, we introduce a numerical scheme that preserves the asymptotic behavior of the discrete velocity kinetic model on general grids. The limiting scheme is schown to be the so-called DDFV scheme [9]. Stability properties of the scheme are studied, and various numerical results illustrate its effeciency.
References
[1] V. Stobiac, P.A. Tanguy and F. Bertrand, 2013. Boundary Condition Strategies for Lattice Boltzmann Simulations: Application to Industrial Mixing Flows, Computers and Fluids, 73, 145-161
[2] D. Vidal, C. Ridgway, G. Pianet, J. Schoelkopf, R. Roy and F. Bertrand, 2009. Eect of Particle Size Distribution and Packing Compression on Fluid Permeability as Predicted by Lattice-Boltzmann Simulations, Comp. Chem. Eng., 33, 256-266
[3] M. Junk, A. Klar, L.-S. Luo, Asymptotic analysis of the Lattice Boltzmann Equation, J. Comput. Phys, 210, 2005, pp. 676{704
[4] S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, in Proceedings of the 5th summer school on "Methods and Models of Kinetic Theory", Porto Ercole (Grosseto, Italy), Rivista di Matematica dell'Universita di Parma, 3, 2012, pp. 177{216
[5] E. Franck, Design and numerical analysis of asymptotic preserving schemes on unstructured meshes. Application to the linear transport and Friedrichs systems, PhD thesis (in French), Univ. Paris VI, 2012
[6] C. Berthon, P.G. LeFloch, R. Turpault, Late-time/sti relaxation asymptotic-preserving approximations of hyperbolic equations, Math. Comp., 2012
[7] F. Hermeline, A nite volume method for the approximation of diusion operators on distorded meshes, J. Comput. Phys., 160, 2000, pp. 481{499