A Multiple-Relaxation-Time Lattice Boltzmann Model for General Nonlinear Anisotropic Convection–Diffusion Equations

Abstract

In this paper, based on the previous work [B. Shi, Z. Guo, Lattice Boltzmann model for nonlinear convection-diffusion equations, Phys. Rev. E 79 (2009) 016701], we develop a general multiple-relaxation-time (MRT) lattice Boltzmann model for nonlinear anisotropic convection-diffusion equation (NACDE), and show that the NACDE can be recovered correctly from the present model through the Chapman-Enskog analysis. We then test the MRT model through some classic CDEs, and find that the numerical results are in good agreement with analytical solutions or some available results. Besides, the numerical results also show that similar to the single-relaxationtime (SRT) lattice Boltzmann model or so-called BGK model, the present MRT model also has a second-order convergence rate in space. Finally, we also perform a comparative study on the accuracy and stability of the MRT model and BGK model by using two examples. In terms of the accuracy,

A block triple-relaxation-time lattice Boltzmann model for nonlinear anisotropic convection–diffusion equations

“…A number of simulations of isotropic and anisotropic convection-diffusion equations are conducted to validate the present B-TriRT model. The results indicate that the present model has a second-order accuracy in space, and is also more accurate and more stable than some available lattice Boltzmann models.in the study of nonlinear problems, such as reaction-diffusion equation [33,34,35,36], isotropic convection-diffusion equations (CDEs) [37,38,39,40], anisotropic convection-diffusion equations [41,42,43,44,45,46,47,48,49], and some high-order partial differential equations [50,51,52,53].Actually, the most widely used model for nonlinear problems is lattice Bhatnagar-Gross-Krook (LBGK) model due to its high computational efficiency, but it is usually unstable for the convection-dominated problems [4]. To overcome this problem, some improved models have been proposed which can be generally grouped into two major categories: (1) the models through introducing additional parameters; (2) the models through modifying collision operator.…”

“…in the study of nonlinear problems, such as reaction-diffusion equation [33,34,35,36], isotropic convection-diffusion equations (CDEs) [37,38,39,40], anisotropic convection-diffusion equations [41,42,43,44,45,46,47,48,49], and some high-order partial differential equations [50,51,52,53].…”

Discrete unified gas kinetic scheme for nonlinear convection-diffusion equations

“…The analytical solution of Eq. (45) can be given by [21] φ(x, y, t) To test the convergence rate of DUGKS for this problem, some simulations were carried out at time t = 1.0, the lattice sizes are varied from 25 × 25 to 100 × 100, and time step ∆t = 1.0 × 10 −5 . From the results in Fig.…”

“…The Gaussian hill problem is described by the following anisotropic convection diffusion equation [21] ∂ t φ + ∇ · (φu) = ∇ · (K · ∇φ),…”

“…Recently, Chai et al. [21] presented a MRT LB model for general NACDEs without any assumptions on the convection and diffusion terms. Although everything looks perfect, some restrictions still exist in all the above LB models.…”