A finite volume method for two-dimensional Riemann-Liouville space-fractional diffusion equation and its efficient implementation

We develop a finite volume method based on Crank-Nicolson time discretization for the two-dimensional nonsymmetric Riemann-Liouville space-fractional diffusion equation. Stability and convergence are then carefully discussed. We prove that the finite volume scheme is unconditionally stable and convergent with second-order accuracy in time and min{1 + alpha, 1 + beta} order in space with respect to a weighted discrete norm. Here 0 < alpha, beta < 1 are the space-fractional order indexes in xand ydirections, respectively. Furthermore, we rewrite the finite volume scheme into a matrix form and develop a matrix-free preconditioned fast Krylov subspace iterative method, which only requires storage of O(N) and computational cost of O(NlogN) per iteration without losing any accuracy compared to the direct Gaussian elimination method. Here Nis the total number of spatial unknowns. Consequently, the fast finite volume method is particularly suitable for large-scale modeling and simulation. Numerical experiments verify the theoretical results and show strong potential of the fast method. (c) 2019 Elsevier Inc. All rights reserved.