Numerical analysis and fast implementation of a fourth-order difference scheme for two-dimensional space-fractional diffusion equations

摘要

In this paper, a fourth-order difference scheme (FODS) is proposed for solving the two-dimensional Riesz space-fractional diffusion equations with homogeneous Dirichlet boundary conditions. It is proved that the FODS is uniquely solvable, unconditionally stable, and convergent with order O(τ2+hx4+hy4) in the discrete L∞- norm, where τ is the time step size, and hx, hy are the space grid sizes in the x direction and the y direction, respectively. Based on the special structure and symmetric positive definiteness of the coefficient matrix, a fast method is developed for the implementation of the FODS. The fast method reduces the storage requirement of O(N2) and computational cost of O(N3) down to O(M+J) and O(Nlog N), where N=MJ, M and J are the numbers of the spatial grid points in the x direction and the y direction, respectively. Finally, several numerical results are shown to verify the theoretical results and the efficiency of the fast method.