Lattice Boltzmann Method For A Class Of Spatial Fractional Order Equations

In scientific computing and practical applications,solving fractional partial differential equations is an important research content.Many scholars have studied solving fractional partial differential equations.For partial differential equations of the spatial fractional order,generally speaking,the solution methods are mainly divided into two categories,one is analytical solution and the other is numerical solution.Since only a few special equations can be derived from analytical solutions,the application range is not extensive,so the approximate solution of the equation is mainly obtained by numerical methods.From the perspective of the progress made by the lattice Boltzmann method(LBM)and its unique flexibility,it will bring a new idea to solve partial differential equations of the spatial fractional order.This paper mainly applies the lattice Boltzmann method to solve a class of Riesz spatial fractional order partial differential equations,and the article is mainly divided into two parts.In the first part,the one-dimensional Riesz spatial fractional partial differential equation is processed,and three methods for dealing with Riesz spatial fractional derivatives are given,namely Grünwald–Letnikov approximation method,complex rectangle method,and complex trapezoidal method.Then,combined with Taylor expansion and Chapman-Enskog expansion,the D1Q3 lattice Boltzmann model is constructed and the macroscopic equation is restored,the equilibrium distribution function is derived,and then the error analysis is carried out,and the effectiveness of the three algorithms is verified by numerical simulation and finite difference method comparison.In the second part,the two-dimensional Riesz spatial fractional partial differential equation is processed,and the Riesz spatial fractional derivative in both directions is processed separately by the Grünwald–Letnikov approximation method.Then,by Taylor expansion and Chapman-Enskog expansion,the D2Q9 lattice Boltzmann model was constructed and the macroscopic equation was restored,and the equilibrium distribution function was derived,and then the error analysis was carried out,and the effectiveness of the algorithm was verified by numerical simulation.