The Lattice Boltzmann Method For A Class Of Third Order Partial Differential Equations With Variable Coefficients

The lattice Boltzmann method is a new method of computational fluid dynamics.Characterized as clear physical background,easy boundary processing,favorable parallelism and extensibility,so lattice Boltzmann method arouses widely concern from scholars.In recent years,many scholars have applied lattice Boltzmann method to the numerical solution of partial differential equations,and have achieved great progress,which makes lattice Boltzmann method become an effective numerical method to solve all kinds of nonlinear partial differential equations.For most of the partial differential equations with variable coefficients,it is difficult to find their analytical solutions.Therefore,researchers try to find the appropriate numerical methods to obtain the numerical solutions of these complex equations.This paper aims to solve the third order partial differential equation with variable coefficients by lattice Boltzmann method.Firstly,a lattice Boltzmann model is established for a class of third order variable coefficient partial differential equations with source term.The compensation function for restoring the convection term and the source term of the macroscopic equation is added to the evolution equation.According to the multi-scale Chapman-Enskog expansion technology,based on the D1Q5 velocity model,the equilibrium distribution function and compensation function are properly selected,and the macroscopic equation with third-order accuracy is finally recovered.Secondly,the stability analysis of the model is given.Finally,different examples are selected for numerical simulation to verify the effectiveness and accuracy of the model established in this paper.The numerical results show that the numerical solution agrees well with the exact solution,and the numerical accuracy is consistent with the theoretical accuracy.Compared with the results of other numerical methods,the results obtained in this paper have higher calculation accuracy and longer evolution time of solutions.The model can also be used to solve the Kd V equation,the modified Kd V equation with variable coefficients,and the Gardner equation,which has a wide range of applications.