Lattice Boltzmann Method For Elastic Wave Equation

The lattice Boltzmann method（LBM） originated from the lattice gasautomat（aLGA）which was proposed in1970s，therefore in this paper，we willintroduce the theoretical basis of the lattice gas automata method at first. TheLGA is a fully discrete dynamical systems: fluid is discretized as a largenumber of particles, the flow field is discretized into a regular lattice, the time isdiscretized into time series with fixed step, in LGA, the particle can only movealong the grid line, and in a time step the particle can only move from onelattice note to the nearest adjacent lattice note，the particle velocity is also a setof discrete and limited velocity.In the lattice Boltzmann model, we use the distribution function of particleinstead of particle itself in LGA to do the evolution, and we wil directly use thelattice Boltzmann equation in the evolution equations, and calculate the fluiddensity and speed according to the distribution function. With the discretespace-time step, we can make the complex Boltzmann integral-differentialequations into simple discrete lattice Boltzmann equation.In the first chapter of this paper we introduce the basic idea of the latticeBoltzmann method in detail as well as some basic models and the currentdevelopment and research.In the he second chapter，at first we introduce the elastic wave equation,then the lattice Boltzmann equation is applied to the elastic wave equation.Firstly apply the Taylor expansion to the lattice Boltzmann equation，secondly apply Chapman-Enskog expansion to distribution functions f_α（x|�',t）、additional distribution function Ω’_α（x|�', t）and the time t, then do the multiscaleexpansion analysis and then according to the law of conservation ofmacroscopic definition condition, we can get a series of partial differentialequations of equilibrium distribution function on different scales.In the third chapter, according to the partial differential equations of theequilibrium distribution function，we can recover the dynamic equation，and according to the characteristics of the elastic wave equation, and with thesecond-order source term, through the appropriate definition of high-ordermoment expressions, we can get the modified elastic wave equation correctly.And then through the modified wave equation，we can get the truncation errorof the equation and know its form, and judge the accuracy of the truncationerror, and obtain the equilibrium distribution function f_α^eq（x|�', t）according to theconservation condition.