Alternating Direction Implicit ParallelAlgorithms For Fractional Diffusion Equations

We are familiar with classic diffusion equations,but fractional differential diffusion equations have a wider range of applications.In this article we will introduce rapid method for solving three-dimensional space-fractional diffusion equations whose form is as follows:Unlike integer order differential solution of diffusion equations,we use Meerschaert-Tadjeran finite difference method for(0.1).Then the finite differ-ence method can be expressed in the matrix form (0.2)is unconditionally stable and first-order convergent [31].Fractional differential operators are non-locality and differential operator coefficient matrixes of one-dimensional problems are usually full arrays.High-dimensional problems’ coefficient matrix sparsity is poor and storage require-ments are too high.The heavy workload of calculation and low computational efficiency severely restricts the promotion and application of fractional partial differential equations.Inspired by alternating direction implicit finite difference methods for integer-order partial differential equations,the authors[Wang,Du]propose al-ternating direction finite difference method for three-dimensional fractional dif-fusion equations and use FFT(Fast Fourier Transform)to solve one-dimensional problems in each direction and effectively improve computational efficiency.Given the importance of the numerical calculations of fractional partial differential equations and based on the results of earlier successions,we try to conduct parallel algorithms of alternating direction finite difference method for space fractional diffusion equation,which are based on message passing interface(MPI,Message Passing Interface) in this article.In accordance with alternating direction "direction"order,we conduct par-allel algorithms of calculation in x direction first and then conduct parallel algorithms of calculation in y direction and z direction.Also inherited by literature [27],we use FFT to solve one-dimensional problems by conjugate gradient method.The article is divided into three chapters:Chapter one briefly introduces parallel computing research contents and parallel principles;Chapter two introduces fractional history and its several definitions,out-line the space-fractional equation we attempt to solve and present the corre-sponding finite difference method,prove the unconditional stability and con-vergence rate of the ADI formulation.In the end of this chapter,we introduce a fast algorithm [25,26,27,28,30,31]based on the coefficient matrix of spe-cial transformation,which means we convert the Toeplitz matrix into a cyclic matrix,then we use FFT to the cyclic matrix,simplify the coefficient and solve one-dimensional problems by conjugate gradient method;Chapter three is mainly about the specific implementation steps of alter-nating direction parallel algorithms,conduct numerical experiments to exam-ine the performance of parallel efficiency of fast parallel algorithm.