Application Of Lattice Boltzmann Method Based On The D1Q5 Velocity Model To Numerically Solve Partial Differential Equations

Lots of partial differential equations are involved in the fields of science and engineer-ing applications.With the rapid development of computer technology and computational mathematics,numerical methods for partial differential equations come out more and more.The lattice Boltzmann method has rapidly grown into a popular numerical method with its unique advantages,and it is widely used in the numerical solutions of nonlinear partial d-ifferential equations.Based on the works by scholars,we investigate and apply the lattice Boltzmann method based on the D1Q5 velocity model to numerically solve two types of par-tial differential equations,including the generalized Boussinesq equation and fourth–order partial differential equations with variable coefficients.Our fundamental work is as follows:1.A general propagation lattice Boltzmann model for the generalized Boussinesq e-quation is developed.Different local equilibrium distribution functions are selected,and the macroscopic equation is correctly recovered by Chapman–Enskog multi–scale analysis and Taylor expansion technique.In order to verify the validity of the model,some generalized Boussinesq equations with initial boundary value problems are numerically solved.2.We propose a unified lattice Boltzmann model for the fourth–order partial differential equations with time–dependent variable coefficients.The equilibrium distribution functions f_k~0 and the compensation functions g_k have unified forms,which avoids the need of dif-ferent models building for different partial differential equations.The truncation error of the model is reduced by adding a compensation function to the evolution equation.The Chapman–Enskog expansion and Taylor expansion methods are used to correctly recover the macroscopic equations.Based on the proposed model and algorithm,some constant co-efficient and variable coefficient partial differential equations are successfully solved.The validity and stability of the model are verified.