The Lattice Boltzmann Method For Convection-diffusion Equation

Abstract:The convection-diffusion equations are a kind of basic Mathe-atical and physical equations,which can be applied in many fields of che-mistry, physics, biology, environmental engineering etc.But Many practi-al problems associated with the convection-diffusion equation is difficult to obtain analytical solutions,so we can only obtain it’s numerical solution by various of numerical methods.Lattice Boltzmann method is a new me-thod of computational fluid dynamics,which has been proposed since19-80s.It takes into account issues from the perspective of mesoscopic,which is a breakthrough and innovation to traditional method.In recenty years,L attice Boltzmann method has made successful appliccations both in solve partial differential equations and engineering simulations.Firstly,In view of one-dimensional convection diffusion without sou-rce term,based on D1Q2discrete velocity model and Chapman-Enskog multi-scale analysis,we rewrite the D1Q2lattice model’s evolution equa-ion as a three levels difference scheme of macro quantity.Then we imple-ments two numerical examples,the arrived numerical solutions are in agr-eement with exact solutions,which is proved that The Lattice Model for partial differential equations is a special kind of difference schemes in so-me sense.Secondly,In view of one-dimensional convection diffusion without source term,basing on D2Q4Lattice velocity model we deduces the cond-itions of equilibrium distribution function must to be satisfied in every v-elocity directions and gives the specific expressions of the equilibrium di-stribution function.Through Chapman-Enskog multi-scale analysis techn-ology, taking the time scale directly to the second order and the spatial sc-ale to the first order, the equilibrium distribution function can recover the original convection-diffusion equation, thus the new D2Q4Lattice Boltz-mann(LB) model for solving two-dimensional convection diffusion equa-tion is constructed.Then we have implemented a diffusion equation and two convection diffusion equation with different initial and boundary co-nditions, the numerical results are in good agreement with analytic soluti-ons.Furthermore,the boundary error is very low compared with related d- ocument, therefore the effectiveness is verified.Finally,we study the Lattice Boltzmann method for convection diffu-sion equation with source term.By Drawing relevant paper’s treatment of source term,we adapt the LBM evolution equation to increase the source term treatment so that we can recover the original equation on the basis of the LBM model we have arrived before.In addition,the arrived LBM mo-del keep with the equilibrium distribution function unchanged.Afterwards we solved one and two convection-diffusion equation respectively.The n-umerical results shows the effectiveness of the our LBM model with sou-rce terms.