A higher-order moment method of the lattice Boltzmann model for the conservation law equation

Abstract In this paper, we proposed a higher-order moment method in the lattice Boltzmann model for the conservation law equation. In contrast to the lattice Bhatnagar–Gross–Krook (BGK) model, the higher-order moment method has a wide flexibility to select equilibrium distribution function. This method is based on so-called a series of partial differential equations obtained by using multi-scale technique and Chapman–Enskog expansion. According to Hirt’s heuristic stability theory, the stability of the scheme can be controlled by modulating some special moments to design the third-order dispersion term and the fourth-order dissipation term. As results, the conservation law equation is recovered with higher-order truncation error. The numerical examples show the higher-order moment method can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the conservation law equation.

[1]  Luo Li-Shi,et al.  Theory of the lattice Boltzmann method: Lattice Boltzmann models for non-ideal gases , 2001 .

[2]  Robert S. Bernard,et al.  Boundary conditions for the lattice Boltzmann method , 1996 .

[3]  Skordos,et al.  Initial and boundary conditions for the lattice Boltzmann method. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[4]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[5]  Yeomans,et al.  Lattice Boltzmann simulation of nonideal fluids. , 1995, Physical review letters.

[6]  Guangwu Yan,et al.  A multi-energy-level lattice Boltzmann model for the compressible Navier-Stokes equations , 2007 .

[7]  Guangwu Yan,et al.  An implicit Lagrangian lattice Boltzmann method for the compressible flows , 2006 .

[8]  D. d'Humières,et al.  Multiple–relaxation–time lattice Boltzmann models in three dimensions , 2002, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[9]  P. Lallemand,et al.  Theory of the lattice boltzmann method: dispersion, dissipation, isotropy, galilean invariance, and stability , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  John Abraham,et al.  Three-dimensional multi-relaxation time (MRT) lattice-Boltzmann models for multiphase flow , 2007, J. Comput. Phys..

[11]  O. Filippova,et al.  Lattice-Boltzmann simulation of gas-particle flow in filters , 1997 .

[12]  Zhenhua Chai,et al.  A novel lattice Boltzmann model for the Poisson equation , 2008 .

[13]  R. Benzi,et al.  The lattice Boltzmann equation: theory and applications , 1992 .

[14]  Guang-wu Yan,et al.  Recovery of the Solitons Using a Lattice Boltzmann Model , 1999 .

[15]  Chen Yaosong,et al.  Simple lattice Boltzmann model for simulating flows with shock wave , 1999 .

[16]  John G. Georgiadis,et al.  Migration of a van der Waals bubble: Lattice Boltzmann formulation , 2001 .

[17]  A. Ladd.  Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation , 1993, Journal of Fluid Mechanics.

[18]  Irina Ginzburg,et al.  Variably saturated flow described with the anisotropic Lattice Boltzmann methods , 2006 .

[19]  Y. Pomeau,et al.  Lattice-gas automata for the Navier-Stokes equation. , 1986, Physical review letters.

[20]  S Succi,et al.  Ground-state computation of Bose-Einstein condensates by an imaginary-time quantum lattice Boltzmann scheme. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  I. Ginzburg.  Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation , 2005 .

[22]  Michihisa Tsutahara,et al.  Lattice Boltzmann method for the compressible Euler equations. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Ping Dong,et al.  Lattice Boltzmann schemes for the nonlinear Schrödinger equation. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Guangwu Yan,et al.  Lattice Bhatnagar—Gross—Krook model for the Lorenz attractor , 2001 .

[25]  T. G. Cowling,et al.  The mathematical theory of non-uniform gases , 1939 .

[26]  R. J. Mason,et al.  A Multi-Speed Compressible Lattice-Boltzmann Model , 2002 .

[27]  Yeomans,et al.  Lattice Boltzmann simulations of liquid-gas and binary fluid systems. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[28]  S Succi,et al.  Numerical validation of the quantum lattice Boltzmann scheme in two and three dimensions. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  S. Zaleski,et al.  Lattice Boltzmann model of immiscible fluids. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[30]  E. Tadmor,et al.  Non-oscillatory central differencing for hyperbolic conservation laws , 1990 .

[31]  Uriel Frisch,et al.  Relation between the lattice Boltzmann equation and the Navier-Stokes equations , 1991 .

[32]  A. Harten.  ENO schemes with subcell resolution , 1989 .

[33]  Sauro Succi,et al.  Nonlinear Stability of Compressible Thermal Lattice BGK Models , 1999, SIAM J. Sci. Comput..

[34]  R. Mason.  A Compressible Lattice-Boltzmann Model , 2000 .

[35]  Shiyi Chen,et al.  LATTICE BOLTZMANN METHOD FOR FLUID FLOWS , 2001 .

[36]  Yang Guangwu.  A Lattice Boltzmann Equation for Waves , 2000 .

[37]  阎广武,et al.  A lattice Boltzmann method for KDV equation , 1998 .

[38]  Shiyi Chen,et al.  Lattice Boltzmann computations for reaction‐diffusion equations , 1993 .

[39]  A. Ladd.  Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results , 1993, Journal of Fluid Mechanics.

[40]  Succi.  Numerical solution of the Schrödinger equation using discrete kinetic theory. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[41]  Huanan Yang,et al.  An artificial compression method for ENO schemes - The slope modification method. [essentially nonoscillatory , 1990 .

[42]  Shan,et al.  Lattice Boltzmann model for simulating flows with multiple phases and components. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[43]  M. Tsutahara,et al.  Lattice Boltzmann model for the compressible Navier-Stokes equations with flexible specific-heat ratio. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  David R. Noble,et al.  Truncation error analysis of lattice Boltzmann methods , 2004 .

[45]  Y. Qian,et al.  Lattice BGK Models for Navier-Stokes Equation , 1992 .

[46]  Chen,et al.  Lattice Boltzmann model for compressible fluids , 1992 .

[47]  Matthaeus,et al.  Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[48]  C. W. Hirt.  Heuristic stability theory for finite-difference equations☆ , 1968 .

[49]  Chenghai Sun,et al.  Lattice-Boltzmann models for high speed flows , 1998 .

[50]  J. Zhou.  Lattice Boltzmann Methods for Shallow Water Flows , 2003 .

[51]  Sauro Succi.  Lattice Quantum Mechanics: An Application to Bose–Einstein Condensation , 1998 .

[52]  Shan,et al.  Simulation of nonideal gases and liquid-gas phase transitions by the lattice Boltzmann equation. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[53]  Pierre Lallemand,et al.  Lattice Gas Hydrodynamics in Two and Three Dimensions , 1987, Complex Syst..

[54]  S. Succi,et al.  Three-Dimensional Flows in Complex Geometries with the Lattice Boltzmann Method , 1989 .

[55]  W. Jin,et al.  A new approach to design high-order schemes , 2001 .