Chen Ying, Liu Xi, Chai Zhenhua, Shi Baochang
School of Mathematics and Statistics, <a href="https://ror.org/00p991c53">Huazhong University of Science and Technology</a>, Wuhan 430074, China; Institute of Interdisciplinary Research for Mathematics and Applied Science, <a href="https://ror.org/00p991c53">Huazhong University of Science and Technology</a>, Wuhan 430074, China; and Hubei Key Laboratory of Engineering Modeling and Scientific Computing, <a href="https://ror.org/00p991c53">Huazhong University of Science and Technology</a>, Wuhan 430074, China.
Phys Rev E. 2024 Jun;109(6-2):065305. doi: 10.1103/PhysRevE.109.065305.
In this paper we first present the general propagation multiple-relaxation-time lattice Boltzmann (GPMRT-LB) model and obtain the corresponding macroscopic finite-difference (GPMFD) scheme on conservative moments. Then based on the Maxwell iteration method, we conduct the analysis on the truncation errors and modified equations (MEs) of the GPMRT-LB model and GPMFD scheme at both diffusive and acoustic scalings. For the nonlinear anisotropic convection-diffusion equation (NACDE) and Navier-Stokes equations (NSEs), we also derive the first- and second-order MEs of the GPMRT-LB model and GPMFD scheme. In particular, for the one-dimensional convection-diffusion equation (CDE) with the constant velocity and diffusion coefficient, we can develop a fourth-order GPMRT-LB (F-GPMRT-LB) model and the corresponding fourth-order GPMFD (F-GPMFD) scheme at the diffusive scaling. Finally, three benchmark problems, the Gauss hill problem, the CDE with nonlinear convection and diffusion terms, and the Taylor-Green vortex flow in two-dimensional space, are used to test the GPMRT-LB model and GPMFD scheme, and it is found that the numerical results not only are in good agreement with corresponding analytical solutions, but also have a second-order convergence rate in space. Additionally, a numerical study on one-dimensional CDE also demonstrates that the F-GPMRT-LB model and F-GPMFD scheme can achieve a fourth-order accuracy in space, which is consistent with our theoretical analysis.
在本文中，我们首先提出通用传播多松弛时间格子玻尔兹曼（GPMRT-LB）模型，并在守恒矩上获得相应的宏观有限差分（GPMFD）格式。然后基于麦克斯韦迭代方法，我们在扩散和声学尺度下对GPMRT-LB模型和GPMFD格式的截断误差和修正方程（MEs）进行了分析。对于非线性各向异性对流扩散方程（NACDE）和纳维-斯托克斯方程（NSEs），我们还推导了GPMRT-LB模型和GPMFD格式的一阶和二阶MEs。特别地，对于具有恒定速度和扩散系数的一维对流扩散方程（CDE），我们可以在扩散尺度下开发一个四阶GPMRT-LB（F-GPMRT-LB）模型和相应的四阶GPMFD（F-GPMFD）格式。最后，使用三个基准问题，即高斯丘问题、具有非线性对流和扩散项的CDE以及二维空间中的泰勒-格林涡旋流，来测试GPMRT-LB模型和GPMFD格式，结果发现数值结果不仅与相应的解析解吻合良好，而且在空间上具有二阶收敛率。此外，对一维CDE的数值研究还表明，F-GPMRT-LB模型和F-GPMFD格式在空间上可以达到四阶精度，这与我们的理论分析一致。
