The Lattice Boltzmann Method For Some Nonlinear Partial Differential Equations

In this thesis, we discuss applications of simulating some Nonlinear Partial Differential Equations by the lattice Boltzmann Method. As a new mesoscopic numerical approach, LBM is so different from traditional numerical methods for fluids computation, it is based on the viewpoint of the microcosmic model and the microcosmic kinetics.At the beginning of this thesis, some basis summaries of the LBM are introduced. In Chapter 1, we construct a D1Q4 model for simulating KdV-Burgers equation, by the Chapman-Enskog expansion and the multiple-scale technique, we recover the macroscopic equations correctly and the macroscopic variables are derived also. Then we give three examples in Section 1.3 numerical experiments, simulating results are compared with the analytical solution of the KdV-Burgers equation and they are in excellent agreement with each other. In Chapter 2, we establish an explicit scheme of lattice BGK model for sim-ulating breather waves of the Sine-Gordon equation. Using a modified lattice Boltzmann equation and the D1Q3 model, we obtained Sine-Gordon equation with 2 order truncation error. Simulating results fit close in with the theoretical results. In Chapter 3, we propose a lattice Boltzmann model with an amending function for simulating four-order derivative nonlinear partial differential equations. This model is different from those existing models, because it lets the time step be equivalent to the square of the space step and the solution with 5-order accuracy. The last Chapter, we make a conclusion of the work. In this part, we introduce research progress and achievement of the subject, and point out how to further our research work, such as ideas and suggestions need to be solved.