Research On Lattice Boltzmann Model For Partial Differential Equations

Lattice Boltzmann method (LBM), evolved out of ideas that has been intensely investigated since 1980s. Unlike conventional numerical schemes based on discretizations of macroscopic continuum equations, the lattice Boltzmann method is based on microscopic models and mesoscopic kinetic equations. Its unique micro-particles background makes it show more sta-ble, valid and accurate than conventional numerical schemes. From the date of proposed, the lattice Boltzmann method has been arousing many experts and scholars' interest (such as fluid mechanics, physics, mathe-matics, computer technology) and has become a hot topic research. In the last two decades, the lattice Boltzmann method achieved continuous improvement and rapid development in theory, models, applications and so on.In recent years, many scholars have applied the lattice Boltzmann method in numerical solution of partial differential equations and have made a great progress. With the academic literatures and writings on this subject continuing to emerge, the lattice Boltzmann method has developed into an alternative and promising numerical scheme for numerical solution of partial differential equations. In this paper, we first present an overview of the development process and progress of the lattice Boltzmann method. And we focus on the recent extensions of this method on numerical solution of partial differential equa-tions. We review the modeling principle of the lattice Boltzmann method and elaborate in detail the process of discretization from the continuum Boltzmann equation to lattice Boltzmann equation. We also introduce the basic elements of the lattice BGK model.The main work of this paper is applying the lattice Boltzmann method to solve some partial differential equations numerically. We study in depth the models, algorithms and applications. The fundamental idea of the lattice Boltzmann method is to construct simplified kinetic models that incorporate the essential physics of microscopic or mesoscopic processes so that the macroscopic averaged properties obey the desired macroscopic equations. The lattice Boltzmann method offers many attractive features in solving nonlinear problems.We built two lattice Boltzmann models of the one dimensional non-linear heat conduction equation: The evolution laws of the particle distribution function for the two models are written as Through adding some terms to the evolution equation and selecting the local equilibrium distribution functions and the distribution functions of the source term properly, a lot of additional terms in the recovered macro-scopic equation can be removed. Both models can achieve the third order accuracy. Numerical simulations with the D1Q4 velocity model for the heat conduction equation are performed. And the numerical results agree well with the exact solutions. This confirms the efficiency and accuracy of the two models.An optimized lattice Boltzmann model is proposed to caculate the generalized Boussinesq equation: In order to eliminate the fourth order truncation error in traditional model, a source term is introduced to the evolution equation and the resulting extended lattice Boltzmann equation is proposed asThis treatment can recover the macroscopic equation to the fourth order accuracy. We use this model to simulate the motion of the soliton. The results are in good agreement with the exact solutions. The soliton moves along the x axis and maintains the amplitude and shape for a long time. We also consider the two dimensional undamped sine-Gordon equa-tion: Applying the implicit scheme to the local equilibrium distribution function fieq(X,t): Whereθis a parameter. The lattice Boltzmann equation becomes We can recover the macroscopic equation to the second order accuracy. The numerical results agree well with the exact solutions. This confirms the efficiency and accuracy of the lattice Boltzmann model. Then we suc-cessfully implement numerical experiments on single pulse propagation and two pulse collision under different orientations. The particle nature, as known for solitons, persists in these two space dimensional solutions as long as the amplitudes of initial data range in a finite interval, similar to the conditions on the vector Maxwell systems.Finally, we consider the two dimensional Burgers'equations with two variables: We rearrange the structure of equations and treat a part of convection term as source term. A lattice Boltzmann model for nonlinear convection-diffusion equation with source term has been improved to solve the two dimensional Burgers'equations with two variables. The lattice Boltzmann equations of u and v are: Through selecting the local equilibrium distribution functions and the dis-tribution functions of the source term properly we can recover the macro-scopic equations with the second accuracy. Numerical experiments for var-ious values of Reynolds number, computational domain are calculated and validated against exact solutions or other published results. It is concluded that the proposed method performs well.