The Lattice Boltzmann Method For A Class Of Nonlinear Partial Differential Equations

Lattice Boltzmann method(LBM) is an emerging and effective numerical method to simulate complex fluid flows and modeling physics in computational fluid dynamics(CFD). The Lattice Boltzmann method based on microscopic models and mesoscopic kinetic equations is a mesoscopic numerical method, so it has essential differences with traditional numerical methods. Because of the microscopic characteristics, it provides lots of advantages of molecular dynamics, for example it has a clear physical picture and inhered parallelism, and it easy to handle boundary conditions and to achieve the program.As a result, it has been concerned by many domestic and international scholars, and has been widely used in numerous research fields such as magnetic fluid, porous media,reaction-diffusion system, incompressible flows and turbulent flow. Recently the lattice Boltzmann models have been applied to simulate nonlinear partial differential equations(NPDEs). The thesis is studied focusing on the numerical solution of a class of one-dimensional nonlinear partial differential equations based on lattice Boltzmann method. Details are as follows:In chapter 1, the lattice Boltzmann method is review in detail, mainly including the research background of it in computational fluid dynamics and its development history and application status.In chapter 2, the basic theory of LBM is presented. Firstly, introduces some classical lattice Boltzmann models, including lattice BGK models and Guo’s model for the incompressible Navier-Stokes(N-S) equations. Then the correct N-S equations can be deduced by the technique of multi-scale expansion, Chapman-Enskog expansion and Taylor expansion.In chapter 3, a lattice Boltzmann model with amending-function for a class of one-dimensional nonlinear partial differential equations was constructed by D1Q5 model.In addition, the stability of the model is be analyzed. Through some numerical experiments based on the above model, we find that it can be applied to solve KdV-Burgers’ equation,KdV equation, Burgers equation, modified Burgers equation. The numerical results show that and the numerical results are in good agreement with analytical solutions, and its error is smaller than the existing literatures’, which verifies that the lattice Boltzmann model is effective in this paper.In chapter 4, summarize research work. LBM is an effective numerical method forsimulating this class of one-dimensional nonlinear partial differential equations. In the future work, we should try to study and simulate its two or three dimension lattice Boltzmann models for more nonlinear partial differential equations.