Study Of Algorithm For Solving Fractional Diffusion Equations

Fractional differential equation has a very extensive application in the field of mathematics and physics, especially, fractional diffusion equation can be more accurate description of many anomalous diffusion phenomenon, such as simulating the structure of permeability, turbulent flow, groundwater contaminant transport, and chaotic dynamics of classical conservative systems. Therefore, studying of fractional diffusion equation plays an important role in practice and theory. But for the nonlocal property of fractional differential operators, the finite difference scheme, finite element and other numerical methods for fractional diffusion equation always tend to generate full coefficient matrices, which require computational cost of O(N3) and the storage of O(N2) , which N is the number of grid points if we use general Gaussian elimination method. There is no doubt that it will increase the complexity of computing and storage space. So, looking for a new fast and efficient numerical algorithm has become a focus of attention.To further improve the performability and stability of Krylov subspace methods, preconditioning technology emerges at a historic moment. preconditioning technology has gradually become the effective method to solve the challenging and complex problems of large-scale computing. In recent years, researchers have found many efficient preconditioners, and the preconditioning technology has become an active research direction comparing to the direct methods and Krylov subspace methods. In terms of numerical solvers for fractional diffusion equation, researchers constructed a lot of classical preconditioners aiming at the discretized matrix, which has a good feature, i.e., it can be written as a sum of diagonal-multiply-Toeplitz matrices. This dissertation mainly introduces three preconditioners. Preconditioned CGNR(PCGNR) with a circulant preconditioner is proposed by Lei and Sun, which is based on the CGNR algorithm. PCGNR algorithm has greatly accelerated the computing speed. Based on the idea of matrix splitting, Lin proposed a new preconditioner, which is a banded matrix, combined with the restarted GMRES in Krylov subspace. This method is called PGMRES, it is a method of fast solving fractional diffusion equation. A k-step ploynomial preconditioner is designed based on the circulant and skew-circulant splitting(CSCS) iteration method proposed by Gu. The ploynomial preconditioner isvery efficient for solving the non-Hermitian Toeplitz systems. These numerical methods with preconditions improved the operation efficency, and preserved O(Nlog N) complexity for Toeplitz system. Moreover, Storage requirement is siginificantly reduced from O(N~2) to O(N).In order to further accelerate the speed of the fractional diffusion equation, improve the accuracy of the numerical approximation, save computation and storage space. The dissertation has conducted the through careful research to fast numerical algorithm for solving fractional diffusion equation. We introduce a permutation matrix to equivalent transform the original nonsymmetric linear system into a symmetric one so as to improve the performance of iterative methods. Our preconditioner avoids tediouscal calculation by the traditional matrix-vector product. We successfully make full use of the Toeplitz structure, as we know, the Toeplitz matrix-vector product can be computed in O(Nlog N) by the Fast Fourier transform. Through the numerical experiments, we can see that the permutation preconditioning proposed in this paper is more superior on the whole performance comparing with the methods with no preconditioning. Our method has faster convergence speed, less calculation and save a lot of time than several methods mentioned in this article.