Fast and efficient finite difference method for the distributed-order diffusion equation based on the staggered grids

Abstract

In this article, an efficient block-centered finite difference scheme is constructed to solve the distributed-order diffusion equation with Neumann boundary condition on the rectangular grids. We use the mid-rectangle formula to approximate the integral term and a fast technique to evaluate the fractional derivative to derive a valid and efficient scheme. The stability and convergence of the efficient scheme for both pressure and velocity are proved rigorously. Finally, we carry out some numerical examples to verify our theoretical analysis and present the efficiency of our scheme.