A unified fourth-order Bhatnagar-Gross-Krook lattice Boltzmann model for high-dimensional linear hyperbolic equations

Abstract

In this work, we first develop a unified Bhatnagar-Gross-Krook lattice Boltzmann (BGK-LB) model for the $d$($d\geq 1$)-dimensional linear hyperbolic equation (L-HE), where the natural moments and the D$d$Q$(2d^2+1)$ [($2d^2+1$) discrete velocities in $d$-dimensional space] lattice structure are considered. Subsequently, at the acoustic scaling, we conduct an accuracy analysis on the developed BGK-LB model by the direct Taylor expansion (DTE) method, and present the second- and third-order moments of the equilibrium distribution functions (EDFs) to ensure that the BGK-LB model can be fourth-order consistent with the L-HE. And on this basis, when considering the Dirichlet boundary condition, the fourth-order full-way and half-way boundary schemes are proposed to approximate the unknown distribution functions to ensure that the BGK-LB model can be overall fourth-order accurate. Thereafter, based on the kinetic entropy theory, we derive the conditions that the fourth-order moments of the EDFs should satisfy to ensure the microscopic entropy stability of the BGK-LB model. In addition, with the aid of the von Neumann stability analysis, we also discuss the $L^2$ stability of the BGK-LB model and numerically plot the stability regions. In particular, from a numerical perspective, we find that the region of microscopic entropy stability is identical to that of $L^2$ stability. Finally, we carry out some numerical experiments to test the accuracy and stability of the BGK-LB model, and the numerical results are in agreement with our theoretical analysis. In addition, we compare the developed full-way and half-way boundary schemes for the Dirichlet boundary condition, which shows that the full-way boundary scheme is more stable.

Figures (13)

• Figure 1: The dotted straight line is the lattice line and the solid straight line is the boundary. White circles (◦) are the lattice nodes in the computational domain, the black circle (•) is the intersection of the boundary and the lattice line. $\vartheta=0$ and $\vartheta=1$ represent the full-way and half-way cases, respectively.
• Figure 2: The microscopic entropy stability region of the BGK-LB model for the three-dimensional L-HE.
• Figure 3: The $L^2$ stability region of the BGK-LB model for the two-dimensional L-HE ($\bm{\xi}\neq \mathbf{0}$).
• Figure 4: The $L^2$ stability region of the BGK-LB model for the three-dimensional L-HE ($\bm{\xi}\neq \mathbf{0}$).
• Figure 5: The convergence rate of the BGK-LB model under the transport velocity $\mathbf{u}=(0.1,0.1)$.