The Lattice Boltzmann Method For Conservative Law Partial Differential Equations

In recent years, computational methods based on the lattice Boltzmann method (LBM) have attracted much attention. They have been developed as an alternative method for computational fluid dynamics (CFD). The applications of the lattice Boltzmann method draw the attention of many scholars. In the thesis we focus on the lattice Boltzmann model for the conservation law equations.Chapter 1 reviewed brieflythe history of the lattice Boltzmann methods.These lattice Boltzmann methods originated from a Boolean fluid model known as the lattice gas automata(LGA) originally developed to overcome certain drawbacks such as the presence of statistical noise and lack of Galilean invariance of LGA for modeling fluid flow based upon kinetictheory. In this chapter, not only the basic models and boundary conditions are been introduced, but some fields of the applications of the LBM are been reviewed.In chapter 2,with the Chapman-Enskog expansion and multi scales technique,we obtain the series of partial differential equations in different time scales in the lattice Boltzmann model as following. The series of equations are useful to construct the lattice Boltzmann models with high-order truncation error for conservation equations. And they can be be applicable for the models in 1D.2D and 3D. The coefficients called Chapman polynomials in this thesis, i.e.C₁,C₂,…,C₆, are similar to the first six Bernoulli polynomials. These Chapman polynomials are been shown as follows, They can be used to indicate coefficients of the dispersion term and the dissipation term to the modified conservation law equation. In order to obtain the conservation laws in different time scales, we need supply those partial differentials on time t_i to the equilibrium distribution function. Then the series of equations are changed into the following forms: In chapter 3, with the series of equations, we give some lattice Boltzmann models for conversation laws equations. We find that the relations of the higher-order moments of the equilibrium distribution functions and the flux F(u) have to meets some requirements in our model. The relations of the higher-order moments of the equilibrium distribution functions and the flux F(u) can be expressed as follows: In LBM model for 1D conservation law equation, if the higher-order momentsmeet the requirements, we can construct a higher-order accuracy model. The numerical examples show the higher-order moment can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the conservation law equation. In this model, we obtain the fifth dispersionterm and the sixth dissipation term, and discuss their effect according the Hirt's heuristic stability.In chapter 4, we discussed the lattice Boltzmann method for Euler equations.We recover the Euler equations with the series of equations, and discuss the modeling of the LBM for the Euler equations in a general way. We propose a multi-energy-level lattice Boltzmann model for Euler equationsand give the numerical examples of the model.In chapter 5,we propose a Lagrangian lattice Boltzmann method for simulatingthe Euler equations. The new scheme simulates fluid flows based on Lagrangian coordinate. The scheme is applicable for the flows with shock waves and contact discontinuities. The Lagrangian lattice Boltzmann model is a simpler version than the corresponding Eulerian coordinates, because the convection term of the Euler equations disappears. The numerical simulationsconform to classical results.