The Lattice Boltzmann Model Of Improved Boussinesq Equation

Lattice Boltzmann method (LBM) is a new method for modeling fluid system and simulation in recent 20 years in the world. With its rapid development, LBM has become a fluid calculation method that is different from classical numerical simulation methods. It is a new science calculation method based on the microscopic model and the kinetic equation of mesoscopic. Unlike traditional numerical methods which directly compute the partial differential equations to get the desired physical quantities, the LBM is to solve the discrete velocity Boltzmann equation based on the particle distribution function, which describes the microscopic picture of particle movement from the statistical physics point of view. Then the macroscopic physical quantities, such as density and velocity, are derived with a sum or moment integrations of the distribution function. This background brings LBM a number of advantages,such as algorithmic simplicity, fully parallel computation and easy implementation of complex boundary conditions. Up to date, LBM has not only displayed well in modeling fluid flows, but also showed the strong ability to simulate nonlinear mathematical-physical equations. Nevertheless, only a small part of partial differential equations are solved through the lattice Boltzmann model. Many complex partial differential equations, such as with complex differential items, non-linear source terms, etc., there are still some unresolved issues in the model structure, numerical accuracy and numerical stability.In this paper, we will study Improved Boussinesq equation and propose a Numerical solution model by LBM. The first chapter briefly introduces on the historical background and features of the LBM. And we describe the current status of the process and prospects of research at home and abroad. The second chapter summarized the basic theory of the lattice Boltzmann method. We introduce the birth and development of the method, and list several commonly used one-dimensional and two-dimensional model of a single relaxation time. Then, we list the common boundary treatment methods. In the third chapter introduces, we built several lattice Boltzmann models for the Improved Boussinesq equation. And then we study the characteristics of the models. The fourth chapter is numerical simulation. First, there is the problem by calculating the analytic solution to verify the actual accuracy and validity of the model. Then we will verify the validity and scope through other examples. The fifth chapter of this paper are summarized and given the prospect of further work.