Research And Application Of Two-Relaxation-Time Lattice Boltzmann Model For Convection-Diffusion Equation

Convection-diffusion equation is an important branch of partial differential equations and is widely used in fluid mechanics,energy development,environmental science,and electronic science.The mesoscopic lattice Boltzmann(LB)method has the advantages of simple algorithm,easy handling of complex boundaries and intrinsic parallelism,which makes it a great advantage in solving convection-diffusion equations.This paper is based on the two-relaxation-time(TRT)LB model and has carried out a series of studies on the general convection diffusion equation.Firstly,the original TRT model is simplified in form.The simplified TRT model can be directly seen as an extension of the single-relaxation-time(SRT)LB model,and can also be verified as a special case of the standard multiple-relaxation-time(MRT)LB model.In addition,through the Chapman-Enskog analysis,the current model can be accurately restored to the general convection-diffusion equation.Secondly,the TRT model is extended to the complex-valued convection-diffusion equation,and theoretical analysis of the TRT model is carried out.The theoretical analysis includesa)analyzing the eigenvectors of the collision matrix of the TRT model,and obtains the connection between it and the transformation matrix of the general MRT model.b)the unified form of the collision matrix of the TRT model DdQ2 n and DdQ2n+1 is obtained.c)the stability analysis of the free relaxation parameter of the current model to determine its stability region.Finally,TRT model is applied to two-dimensional linear convection-diffusion equation,three-dimensional linear convection-diffusion equation,four-dimensional linear convection-diffusion equation,nonlinear heat conduction equation,the complex-valued nonlinear Schr?dinger equation and complex Ginzburg-Landau equation.The numerical results show that the stability region of the free relaxation parameters of the TRT model is consistent with our theoretical analysis.In addition,by comparing the results with the SRT model,we found that the current model is significantly better than the SRT model in terms of stability and accuracy.