Lattice Boltzmann Model for High-Order Nonlinear Partial Differential Equations

TL;DR

This paper introduces a versatile lattice Boltzmann model capable of solving high-order nonlinear PDEs, demonstrating its effectiveness through simulations of various complex equations with results matching analytical and existing numerical solutions.

Contribution

A novel lattice Boltzmann framework for high-order nonlinear PDEs using auxiliary moments for accurate macroscopic recovery.

Findings

Model accurately solves high-order nonlinear PDEs.

Numerical results agree with analytical solutions.

Applicable to a range of complex nonlinear equations.

Abstract

A general lattice Boltzmann (LB) model is proposed for solving nonlinear partial differential equations with the form $\partial_t \phi+\sum_{k=1}^{m} \alpha_k \partial_x^k \Pi_k (\phi)=0$, where $\alpha_k$ are constant coefficients, and $\Pi_k (\phi)$ are the known differential functions of $\phi$, $1\leq k\leq m \leq 6$. 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.