A fast solver for multidimensional time–space fractional diffusion equation with variable coefficients

Abstract

In this paper, we study a discretization scheme and the corresponding fast solver for multi-dimensional time–space fractional diffusion equation with variable coefficients, in which L1 formula and shifted Grünwald formula are employed to discretize the temporal and spatial derivatives, respectively. A divide-and-conquer strategy is applied to the large linear system assembling discrete equations of all time levels, which in turn requires to solve a series of multidimensional linear systems related to the spatial discretization. Preconditioned generalized minimal residual method is employed to solve the spatial linear systems resulting from the spatial discretization. The discretization is proven to be unconditionally stable and convergent in the sense of infinity norm for general nonnegative coefficients. Numerical results are reported to show the efficiency of the proposed method.