Lattice Boltzmann Model for High-Order Nonlinear Partial Differential Equations

Abstract

A general lattice Boltzmann (LB) model is proposed for solving nonlinear partial differential equations with the form ${\partial }_t\varphi +{\sum }_{k=1}^m{\alpha }_k{\partial }_x^k{\Pi }_k(\varphi )=0$, where ${\alpha }_k$ are constant coefficients, and ${\Pi }_k(\varphi )$ are the known differential functions of $\varphi$, $1\le k\le m\le 6$. The model can be applied to the common nonlinear evolutionary equations, such as (m)KdV equation, KdV-Burgers equation, K($m,n$) equation, Kuramoto-Sivashinsky equation, and Kawahara equation, etc. Unlike the existing LB models, the correct constraints on moments of equilibrium distribution function in the proposed model are given by choosing suitable \emph{auxiliary-moments}, and how to exactly recover the macroscopic equations through Chapman-Enskog expansion is discussed in this paper. Detailed simulations of these equations are performed, and it is found that the numerical results agree well with the analytical solutions and the numerical solutions reported in previous studies.

18 pages, 4 figures

References in corpus (3)

• Lattice Boltzmann methods for multiphase flow and phase-change heat transfer
• Factorization symmetry in lattice Boltzmann simulations
• A family of single-node second-order boundary schemes for the lattice Boltzmann method

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• Discrete effects on some boundary schemes of multiple-relaxation-time lattice Boltzmann model for convection-diffusion equations
• A Structured Approach to the Construction of Stable Linear Lattice-Boltzmann Collision Operators
• A general multiple-relaxation-time lattice Boltzmann model for nonlinear anisotropic convection-diffusion equations
• Efficient lattice Boltzmann models for the Kuramoto-Sivashinsky equation