Lattice Boltzmann Method For A Class Of Patrial Differential Equations With Variable Coefficient

As a novel numerical tool for the computational fluid dynamics, lat-tice Boltzmann method (LBM) has essential diferences with conventionalnumerical methods. LBM is a mesoscopic numerical method based onmicroscopic models and mesoscopic kinetic equations. The microscopiccharacteristics provide many of advantages, for example its physical pic-tures are clear and it is easy for programming, intrinsical parallel, and tohandle complex boundary conditions. So LBM was closely watched byresearchers home and abroad. It has been widely used in various fields, es-pecially in the areas where many of the conventional numerical methods arenot competent. LBM has successfully revealed the mechanism of compli-cated problems, for example porous media, magnetic fluid, biofluid, crystalgrowth, combustion and so on. In recent years, many scholars have appliedthe lattice Boltzmann method in numerical solution of partial diferentialequations and have made a great progress.All is well known that partial diferential equation with variable co-efcient is more complex and can describe complicated physic process innature more profoundly than that with constant coefcient. So it is realis-tically significant to research the partial diferential equation with variable coefcient. The main objective of this paper is to present lattice Boltzmannmodels for the numerical solution of some partial diferential equations withvariable coefcient.This paper firstly summarizes the development process of LBM, andthe major application in numerical solution of partial diferential equationof it. The basic structure of lattice Boltzmann method is also introduced.Secondly, we reviews that LBGK equation is derived from discretization oftime and space for the Boltzmann BGK equation, and then using multi-scale Chapman-Enskog expansion, macroscopic Navier-Stokes equation isrecovered. The famous LBGK equation has the following form:where f_a(x, t) and feqa(x, t) are the local particle distribution function andequilibrium distribution function at time t and position x, respectively.{e0,...,e_b-1}is the discrete velocity set, and t is the time step. τ is thedimensionless relaxation time. The evolution equation based on Eq.(1) isproposed for a class of variable coefcient partial diferential equation. Itsform can be written as:where h_α(x, t) is the amending function, which plays an important role foramending the terms in the recovered macroscopic equation and eliminatingthe errors.A lattice Boltzmann model for the numerical solution of the Fokker-Planck equation was established using Eq.(2). The one-dimensional non-linear Fokker-Planck equation is in the following form: the two-dimensional nonlinear Fokker-Planck equation is in the followingform:During recovering the macroscopic equation, multiscale Chapman-Enskogexpansion for amending function hα(x, t) is up to the first order. hα(x, t)aims to amend the convection term in the recovered macroscopic equa-tion. The D1Q3and D1Q5models are used for recovering one-dimensionalFokker-Planck equation with the second and third order accuracy respec-tively, and the D2Q9model for two-dimensional Fokker-Planck equationwith the third order accuracy. The efciency and numerical accuracy ofthe present model are validated through several numerical experiments. Inparticular, it is efcient to simulate one-dimensional stochastic processesgoverned by the Fokker-Planck equation. And the numerical results agreewell with the exact solutions.Based on Eq.(2) a lattice Boltzmann model for the numerical solutionof the Black-Scholes equation was proposed. The generalized Black-Scholesequation has the formwith the terminal conditionwhere V (S,t) is the value of the European call (put) option at the assetprice S and at timet. r(S,t)>0and d(S,t) are the risk-free interest rate and the dividend respectively. E is the exercise price, T is the maturitydate, and σ(S,t) is the volatility function of S andt. Eq.(3) can beequivalent to the following partial diferential equation with a source term:Actually, our lattice Boltzmann model is established for Eq.(5). During re-covering the macroscopic equation, multiscale Chapman-Enskog expansionfor amending function hα(x, t) is up to the second order. hα(x, t) is usedfor amending the convection and source terms in the recovered macroscopicequation. The D1Q3model is used for recovering Black-Scholes equationwith the second accuracy. In the numerical experiment, European call op-tion, binary option and Butterfly Spread option are simulated efciently,and the numerical accuracy of the present model is validated. Furthermore,our proposed lattice Boltzmann model is general, which can be applied to aclass of variable coefcient partial diferential equation with a source term.