The Parallel Implement Of Fast Fourier Transform In Fractional Diffusion Equations

In recent years, the development of parallel computing is rapidly with the advent of parallel machines, and the parallel computing makes outstanding contributions in a number of major computing problems that can not be solved by traditional methods of computation in the general computer. The range of the parallel computing is very wide, including architecture of the parallel ma-chine, the design of parallel algorithm, procedures for the parallel preparation, parallel compilation system and the evaluation of parallel performance, and so on. As a cross-disciplinary, parallel computing can be understood as a bridge which connect parallel computing problems with the practical application of parallel machine. Compare to the traditional method on a general computer in solving large-scale problems, parallel computing has some advantages, such as improving calculation speed, saving investment and solving larger problems.Fractional differential equations have been widely used in modeling tur-bulent flow, chaotic dynamics of classical conservative systems, groundwater contaminant transport, and applications in biology, physics, chemistry, and even finance. But for the numerical method to solve fractional diffusion equa-tions, the coefficient matrix is dense, the computational work is O(N3) and memory capacity is O(N2). In order to reduce the computation and memo-ry capacity professor Wang and his colleagues develop a fast finite difference method for fractional diffusion equations. The fast finite difference method only requires storage of O(N) and computational cost of O(Nlog2N) while retaining the same accuracy and approximation property as the regular finite difference method.The goal of this paper is to combine the parallel computing and the fast finite difference method so as to achieve the parallel implement of the fast algorithm. In this way, we can shorten the time of calculation and increase the calculation efficiency.In chapter1, we introduce the background and application of fractional differential equations and how to reduce the computation time of fractional diffusion equations in this paper.In chapter2, we introduce the basic knowledge of parallel computing, in-cluding the concept of parallel computing, the history and system structure of parallel machine, the parallel environment-MPI, the analysis of parallel ef-ficiency.In chapter3, firstly, we study the fast algorithm of fractional diffusion equation. Secondly, we analysis the specific parallel implementation of the fast algorithm of fractional diffusion equation in the MPI environment. Because the implementation process involves the parallel of FFT, finally, we introduce the FFT parallel algorithm.