Research On Lattice Boltzmann Model For Partial Differential Equations

This article studies the Lattice Boltzmann Method in solving partial differential equations.In this article, we apply D2Q9model as an example of this method to solving the Two Dimen-sional Heat Conduction Equation and D1Q4model as an example of this method to solving the KPP Equation, then choose an example respectively to do numerical experiment. The com-parition between the numerical results and the exact solutions shows that the Lattice Boltzmann Method is accurate and reliable.So, in this article, we do some research on the Lattice Boltzmann Method as follows:(1) In Chapter1, we introduce the process of emergence and development of Lattice Boltz-mann method, then apply D2Q9model to this equation, we get a simple expression of the equi-librium equation by applying the Chapman-Enskog expansion and the multi-scale technology, describe its pratical application and significant impact.Meanwhile, this chapter introduces some LBM models that have been established so far such as D1Q3, D2Q9, D2Q7(D refers to the number of dimensions and Q refers to the the total number of particles motion direction).(2) In Chapter2, we introduce the background of the Two Dimensional Heat Conduction Equation.After that, we choose a numerical equation and do some experiment using Matlab.The comparition between the numerical results and the exact solutions shows that the Lattice Boltz-mann Method is accurate and reliable.(3) In Chapter3, we apply D1Q4model to the KPP equation. We can get good accuracy by using Taylor expansion and Chapman-Enskog multi-scale technologies. After that, we do some experiment using Matlab. We can test the accuracy and reliability by the comparison of the numerical results and the exact solutions.