高阶非线性偏微分方程的格子Boltzmann模型
Lattice Boltzmann Model for High-Order Nonlinear Partial Differential Equations
摘要：
提出了一种求解$\partial_t ϕ+\sum_{k=1}^{m} α_k \partial_x^k Π_k (ϕ)=0$形式的非线性偏微分方程的一般格子Boltzmann（LB）模型，其中$α_k$是常系数，$Π_k (ϕ)$是$ϕ$的已知微分函数，$1\leq k\leq m \leq 6$ .该模型可应用于常见的非线性演化方程，如（m）KdV方程、KdV-Burgers方程、K（$m,n$）方程、Kuramoto-Sivashinsky方程和Kawahara方程等。与现有的LB模型不同，该模型通过选择合适的\emph{auxiliary-moments}，并讨论了如何通过Chapman-Enskog展开精确恢复宏观方程。对这些方程进行了详细的数值模拟，发现数值结果与以前的研究中报道的解析解和数值解吻合得很好。
英文摘要：
A general lattice Boltzmann (LB) model is proposed for solving nonlinear partial differential equations with the form $\partial_t ϕ+\sum_{k=1}^{m} α_k \partial_x^k Π_k (ϕ)=0$, where $α_k$ are constant coefficients, and $Π_k (ϕ)$ are the known differential functions of $ϕ$, $1\leq k\leq m \leq 6$. 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.
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