Lattice Boltzmann Method For Reaction-Diffusion Equation And Its Numerical Simulations

In recent two decades, the lattice Boltmann method has become an important means of modeling and simulation for fluid flows. Historially, the lattice Boltzmann method originated from the lattice gas automata. Unlike conventional numerical schemes based on discretizations of macroscopic continuum equations, the lattice Boltzmann method is based on microscopic models and mesoscopic kinetic equations, which provides many of the advantages of molecular dynamics, including clear physical pictures. easy implementation of boundary conditions, and fully parallel algorithms.As with many unique advantages, the lattice Boltzmann method has been widely used to simulate fluid dynamics and linear and nonlinear partial differential equations. In this paper, we employ the lattice Boltzmann method to simulate several classic Reaction-Diffusion systems, as follows:In chapter 1, we reviewed the history and current research of the lattice Boltzmann method.In chapter 2, the Chapman analysis of lattice Boltzmann models for Reaction-Diffusion systems is introduced.In chapter 3, we use the lattice Boltzmann method to simulate the Schlogl system. By selecting the appropriate equilibrium distribution function and distribution function of additional items, we give the one-. two-and three-dimensional simulation results for Schlogl system. three-dimensional CIMA system.In chapter 5. the lattice Boltzmann model for Brusselator is constructed, the pattern formations of two-and three-dimensional Brusselator are introduced. Furthermore, the different parameters on the impact of pattern are discussed. Numerical experiments show that, the result simulates by the lattice Boltzmann method obtains good consistency with the solution obtains by finite difference method.In chapter 6, we simulate the Gray-Scott model by the lattice Boltzmann method. The linear stability of Gray-Scott model is analyzed. The pattern formations with different combinations of parameters F and K are classified. In addition, we draw the pattern formations of two-and three-dimensional Gray-Scott model, and compare with the results obtained by the finite difference sheme. The numerical results indicate that lattice Boltzmann method can be used to simulate the Gray-Scott model.We draw the conclusions in chapter 7.