A Class of Unconditioned Stable 4-Point WSGD Schemes and Fast Iteration Methods for Space Fractional Diffusion Equations

In this paper, we consider high order finite difference schemes and fast algorithms for one- and two-dimensional space fractional diffusion equations (SFDEs). We first derive a class of 4-point weighted and shifted Grünwald difference (4WSGD) operators to approximate Riemann–Liouville fractional derivatives. A class of spatially j th ( j≥ 3 ) order CN-4WSGD schemes for SFDEs with variable coefficients are proposed, where CN stands for Crank-Nicolson. As the coefficient matrix of the resulted linear system is Toeplitz-like but non-diagonally dominant, we theoretically analyze the stability and the accuracy of CN-4WSGD schemes by using the generating function of Toeplitz matrix. Moreover, we establish a diagonal-times-Toeplitz-splitting iteration method to solve the linear system, which is proved to be asymptotic uniformly convergent without imposing any extra conditions. Numerical examples are presented to verify the accuracy of the numerical schemes for both smooth and lower smooth solutions and test the efficiency of the proposed iteration methods.