Stability and accuracy of various difference schemes for the lattice Boltzmann method

We test a second order central difference scheme and a first order upwind scheme for the advection of particles in the lattice Boltzmann method for fluid flow. A diffusion term is added to the Boltzmann equation in order to improve stability when using the second order scheme, this term is equivalent to the Lax--Wendroff scheme for a particular value of the diffusion constant. In contrast to the normal lattice Boltzmann method, we allow a particle Courant number less than one. We test the schemes for stability and accuracy using Taylor--Green vortex and channel flows in three dimensions, finding improved stability for some configurations and no loss in accuracy. Both modifications are expected to remove some spurious lattice invariants. The proposed particle diffusion term may also be used to improve the stability of other Boltzmann based methods that use higher order difference schemes. References F. Nannelli and S. Succi. The lattice Boltzmann equation on irregular lattices. J. Stat. Phys., 68(3-4):401--407, Aug 1992. doi:10.1007/BF01341755. Nato Advanced Research Workshop : Lattice Gas Automata, Theory, Implementation, Simulations, Nice, France, Jun 25-28, 1991. G. W. Peng, H. W. Xi, C. Duncan, and S. H. Chou. Finite volume scheme for the lattice boltzmann method on unstructured meshes. Phys. Rev. E, 59(4):4675--4682, Apr 1999. doi:10.1103/PhysRevE.59.4675. N. Z. Cao, S. Y. Shen, S. Jin, and D. Martinez. Physical symmetry and lattice symmetry in the lattice Boltzmann method. Phys. Rev. E, 55(1, Part a):R21--R24, Jan 1997. doi:10.1103/PhysRevE.55.R21. T. Lee and C. L. Lin. A characteristic Galerkin method for discrete Boltzmann equation. J. Comput. Phys., 171(1):336--356, July 2001. doi:10.1006/jcph.2001.6791. X. Y. He and G. D. Doolen. Lattice Boltzmann method on a curvilinear coordinate system: Vortex shedding behind a circular cylinder. Phys. Rev. E, 56(1, Part a):434--440, Jul 1997. doi:10.1103/PhysRevE.56.434. G. L. Zanetti. Hydrodynamics of lattice-gas automata. Phys. Rev. A, 40(3):1539--1548, Aug 1989. doi:10.1103/PhysRevA.40.1539. U. Frisch, D. d'Humieres, B. Hasslacher, P. Lallemand, Y. Pomeau, and J. P. Rivet. Lattice gas hydrodynamics in two and three dimensions. Complex Systems, 1:649--707, 1987. V. Zecevic, M. P. Kirkpatrick, and S. W. Armfield. The lattice Boltzmann method for turbulent channel flows using graphics processing units. ANZIAM J., 52(0):914--928, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3951. D. Bespalko, A. Pollard, and M. Uddin. Direct numerical simulation of fully-developed turbulent channel flow using the lattice Boltzmann method and analysis of OpenMP scalability. In High Performance Computing Systems and Applications, volume 5976 of Lect. Notes Comput. Sci., pages 1--19. Springer Berlin, 2010. doi:10.1007/978-3-642-12659-8\protect \global \let \OT1\textunderscore \unhbox \voidb@x \kern .06em\vbox {\hrule width.3em}\OT1\textunderscore 1. Z. L. Guo, C. G. Zheng, and B. C. Shi. An extrapolation method for boundary conditions in lattice {Boltzmann} method. Phys. Fluids, 14(6):2007--2010, Jun 2002. doi:10.1063/1.1471914. I. Ginzburg, F. Verhaeghe, and D. d'Humieres. Two-relaxation-time lattice Boltzmann scheme: About parametrization, velocity, pressure and mixed boundary conditions. Commun. Comput. Phys., 3(2):427--478, Feb 2008. D. d'Humieres, I. Ginzburg, M. Krafczyk, P. Lallemand, and L. S. Luo. Multiple-relaxation-time lattice boltzmann models in three dimensions. Philos. Trans. R. Soc. Lond. Ser. A-Math. Phys. Eng. Sci., 360(1792):437--451, Mar 2002. doi:10.1098/rsta.2001.0955. Y. H. Qian. Fractional propagation and the elimination of staggered invariants in lattice-bgk models. Int. J. Mod. Phys. C, 8(4):753--761, Aug 1997. doi:10.1142/S0129183197000643. 6th International Conference on Discrete Models for Fluid Mechanics, Boston Univ, Ctr. Computat. Sci, Boston, MA, Aug 26--28, 1996