Discrete Effect On The Halfway Bounce-back Boundary Condition And Stability Analysis Of Multiple-relaxation-time Lattice Boltzmann Model For Convection-diffusion Equations

The convection-diffusion equation is an important partial differential equation,which has a wide range of applications in the environmental engineering, fluid mechanics, materials science and many other fields. With the rapid development of computer technology, numerical simulation has become an important method for solving this problem. The lattice Boltzmann(LB) method, as a mesoscopic numerical approach,has been developed to be an efficient method over the past decades, and also some distinct characteristics, such as easy programming, simple algorithm and natural parallel,which make it have great advantages in solving partial differential equations. In this paper, we will focus on the multiple-relaxation-time(MRT) LB model for two-dimensional convection-diffusion equations.Firstly, we analyze the discrete effect on the halfway bounce-back(HBB) boundary condition of MRT model, where three different discrete velocity models are considered.We first present a theoretical analysis on the discrete effect of the HBB boundary condition for the simple problems with a parabolic distribution in x or y direction, and a numerical slip proportional to the second-order of lattice spacing is observed at the boundary, which means that the MRT model has a seconder-order convergence rate in space. The theoretical analysis also shows that the numerical slip can be eliminated in the MRT model through tuning the free relaxation parameter corresponding to the second-order moment, while it cannot be removed in the Bhatnagar-Gross-Krook(BGK)model unless the relaxation parameter related to diffusion coefficient is set to be a special value. We then perform some simulations to confirm our theoretical results, and find that he numerical results are consistent with our theoretical analysis.Secondly, the von-Neumann analysis was applied to investigate the stability regions though systematically varying the relaxation parameters for different discrete velocity models. From the results, we can find that the relaxation parameter corresponding to theconserved variable has no influence on stability regions, while other relaxation parameters have a considerable effect. In addition, a comparison between the MRT model and the BGK model is also conducted, and we find that the stability of the MRT model can be improved though selecting proper relaxation parameters.