Research On Lattice Boltzmann Method For A Class Of Coupled Partial Differential Equations

Lattice Boltzmann method is a computational fluid dynamics method based on mesoscopic simulation scale.Different from other traditional numerical methods,it is a discrete method on the macroscopic scale and a continuous method on the microscopic scale.In recent decades,the lattice Boltzmann method has been known as an important numerical computational method for solving nonlinear partial differential equations with the advantages of easy processing of boundary conditions,high parallelism,and easy programming in computation.The Kd V equation is an important class of nonlinear partial differential equations and the coupled Kd V equation can describe the interactions between long waves with different dispersion relations.Considering the inhomogeneity of the medium and boundary,the equation with variable coefficients can be used to describe some complex phenomena.Therefore,in this thesis,we apply the lattice Boltzmann method to solve a class of coupled Kd V equations with variable coefficients.Firstly,in this thesis,for a class of coupled Kd V equations with variable coefficients,a unified lattice Boltzmann model with correction and source terms is established by making appropriate deformations in order to deal with the variable coefficients and coupling problems in the equations.The D1Q5 discrete velocity model is used to recover the macroscopic equations with second-order accuracy by Chapman-Enskog expansion and selecting appropriate parameters to adjust the equilibrium state distribution function and the correction function.Then the stability of the established lattice Boltzmann model is analyzed.Secondly,different parameters and different coupling forms are selected to numerically simulate the coupled Kd V equation with constant coefficients,the coupled Kd V equation with variable coefficients,and the coupled Kd V equation with Hirota-Satsuma,respectively.Finally,the validity of the model in this thesis is verified by the errors at different moments and at different time steps.The numerical results show that the numerical solution obtained by the lattice Boltzmann method agrees well with the exact solution,the numerical accuracy is consistent with the theoretical accuracy,and the error of the model in this thesis is smaller compared with other numerical methods.