Preconditioners For Space Fractional Diffusion Equations Discretized By Generalized Backward Difference Method

In this thesis,we consider the preconditioners for the unsteady-state space-fractional diffusion equations.The problem is first discretized in time by the generalized backward difference method,and then discretized in space by the finite volume method.By combining all time-steps together,we obtain a large scale system of linear equations whose coefficient matrix is in the form a sum of two Kronecker products.The main contributions of this thesis are as follows:(1)We observe that the coefficient matrix consists of two parts,each of which has a special structure.Thus we propose a class of Kronecker product splitting preconditioner which is based on alternating direction technique.Under suitable conditions,we prove that all the eigenvalues of the preconditioned matrix are in a disc centered at(1,0)and of radius ? which is less than 1.Furthermore,we modify the preconditioner in order to improve its computational efficiency.Numerical results show that the new preconditioner has good acceleration effect.(2)By utilizing the structure of coefficient matrix,we construct a kind of diagonal-compensation preconditioner.Circulant matrix approximation is applied to reduce the computational cost.Some theoretical results are given and the numerical experiments are carried out to show the performance of the new preconditioner.