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

Abstract

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+α,1+β} order in space with respect to a weighted discrete norm. Here 0<α,β<1 are the space-fractional order indexes in x and y directions, 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(Nlog⁡N) per iteration without losing any accuracy compared to the direct Gaussian elimination method. Here N is 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.