Lattice Boltzmann Method For Reaction-Diffusion Problems

CA(Cellular Automaton) and LBM(Lattice Boltzmann Method) are considered as the effective tools for studying nonlinear complex system .Firstly, we give the general introduction of CA methods and LBM for studying reaction-diffusion problems , then the LBM, whose variables of reaction term is distribution function ,is introduced .This reaction term is written as follows:R? (f,) = M (I, I)fr (F,], t) ± N7,, (I,], k)fr (F,], Oh (F, k, t)r,jr,p,j,k+^j P,7pq (i, j,k,l)fr (F,j;t)f (F, k,t)fq (F,1,t) r.p.q,jjcl-p, q, r It means the species taking part into this reaction process and affecting the density of thes species (the species s is included). M',N',P means the reaction collision Matrices , which is decided by the speeds of the particles that take part into the reaction process .We testifies that theexpression of R^f will resume to R3(p) when the system is in the state near equilibrium.We aims to study the LBM , the way of probing the reaction-diffusion system without convection(the speed of fluid is zero),so the expressio^'of reaction forms proposed by S.P.Dawson,R,(p)^I5 still used.Firstly,LBM for reaction-diffusion problems should be set up ,in the way of the onedimension lattice model with three speeds. Then ,the one dimension reaction-diffusion equation is derived by the Lattice Boltzmann equation with the multi-scade method:= (r3 -0.5), is the diffusion coefficient of s species .By the linear stable analysis for four reaction-diffusion equations, Schl?gl , Lotka-Volterra, Selkov, l3russelator, we get the parameter ranges, which control the solution of the equations, such as limit cycles and Turing instability solution and so on. Furthermore, we also go into the computer simulation according to the parameter ranges . By comparing the simulation and preditation we prove that LBM is the practicable way for studying the reaction-diffusion problems .The third step is to study the LBM of two dimension reaction-diffusion problems . A square lattice model with five speeds is set up the diffusion coefficient of s species in themodel is D, = ^-(i.-, - 0.5) .Using the square lattice model with five speeds and hexagonal lattice model with seven speeds , we do coniputer simulation and analysis for the above reaction processes. In orderto reduce the boundary effect to the simulation result , the periodic boundary condition is adopted. When the reversal reaction process is neglected completely ,there will be more deviation in the result of simulation .we prove that the LBM is an effective way for reaction-diffusion system near equilibrium.