A higher-order moment method of the lattice Boltzmann model for the Korteweg-de Vries equation

In this paper, a lattice Boltzmann model for the Korteweg-de Vries (KdV) equation with higher-order accuracy of truncation error is presented by using the higher-order moment method. In contrast to the previous lattice Boltzmann model, our method has a wide flexibility to select equilibrium distribution function. The higher-order moment method bases on so-called a series of lattice Boltzmann equation obtained by using multi-scale technique and Chapman-Enskog expansion. We can also control the stability of the scheme by modulating some special moments to design the dispersion term and the dissipation term. The numerical example shows the higher-order moment method can be used to raise the accuracy of truncation error of the lattice Boltzmann scheme.

[1]  Mahir Rasulov,et al.  An efficient numerical method for solving the Korteveg-de Vries equation in a class of discontinuous functions , 1999, Appl. Math. Comput..

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

[3]  D. Korteweg,et al.  XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , 1895 .

[4]  V. Karpman,et al.  RADIATION BY WEAKLY NONLINEAR SHALLOW-WATER SOLITONS DUE TO HIGHER-ORDER DISPERSION , 1998 .

[5]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .

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

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

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

[9]  Andrus Salupere,et al.  On the long-time behaviour of soliton ensembles , 2003, Math. Comput. Simul..

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

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

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

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

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

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

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

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

[18]  Z. Horii,et al.  Mass transport theory for the Toda lattices, dispersive and dissipative , 2005 .

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

[20]  N. Zabusky,et al.  Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States , 1965 .

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

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

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

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

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

[26]  J M Hyman,et al.  Compacton solutions in a class of generalized fifth-order Korteweg-de Vries equations. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  G. Lamb.  Elements of soliton theory , 1980 .

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

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

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

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

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

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

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

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

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