Stability and convergence of a Crank–Nicolson finite volume method for space fractional diffusion equations

Abstract

We analyze a Crank–Nicolson finite volume method (CN-FVM) for the time-dependent two-sided conservative space fractional diffusion equation (of order 2−α). We prove that the proposed method is unconditionally stable in a weighted discrete norm and has a convergence rate of order O(τ2+h1+α), where τ and h are the time step size and spatial mesh size, respectively. In addition, we present a matrix-free preconditioned fast BiCGSTAB solver for the discrete linear algebraic system, which has a linear memory requirement and almost linear computational complexity. Numerical experiments show strong potential of the fast CN-FVM.