A fast high-order compact difference method for the fractal mobile/immobile transport equation

In this paper, we present a high-order compact finite difference scheme for fractal mobile/immobile transport model with a Caputo fractional time derivative. The compact finite difference scheme is stable and convergent with convergence order O(tau(2-alpha) + h(4)) in L-2-norm. Furthermore, because of the non-local property of fractional differential operators, it is necessary to find a fast technique to reduce the computational cost. We develop a fast solution technique which is based on a fast Fourier transform. The computational work will reduce to O(MN log(2) N) while the direct method requires the computational cost of O(MN2), where N = tau(-1) and tau is the size of time step, M = h(-1) and h is the size of space step. Moreover, numerical results are consistent with the theoretical analysis.