Stability and convergence of finite difference method for two-sided space-fractional diffusion equations

In this paper, we study and analyse Crank-Nicolson (CN) temporal discretization with certain spatial difference schemes for one- and two-dimensional two-sided space-fractional diffusion equations (TSFDEs) with variable diffusion coefficients. The stability and convergence of the resulting discretization linear systems for TSFDEs with variable diffusion coefficients are proven by a new technique. That is, under mild assumption, the scheme is unconditionally stable and convergent with O(tau(2)+h(l)) (l >= 1), where tau and h denote the temporal and spatial mesh steps, respectively. Further, we show that several numerical schemes with lth order accuracy from the literature satisfy the required assumption. Numerical examples are implemented to illustrate our theoretical analyses.