A fast collocation approximation to a two-sided variable-order space-fractional diffusion equation and its analysis

摘要

We develop a fast indirect collocation method for a two-sided variable-order spacefractional diffusion equation, which models, e.g., the superdiffusive transport of solute in a heterogeneous porous medium. Due to the impact of the variable fractional order, the stiffness matrix loses the Toeplitz structure that is common in the context of constant-order sFDEs. Consequently, the discrete fast Fourier transform technique or fast convolution quadrature method used in the development of fast methods for constant-order fractional diffusion equations no longer apply. We approximate the stiffness matrix by a combination of Toeplitz-like matrices via an entrywise expansion. We prove that the approximated system is asymptotically consistent with the original problem with only O(N log N) memory and O(N log(2) N) operations are required per iteration for the fast solvers. We prove that the fast numerical solution of the approximated system has the same order of accuracy as that of the underlying collocation method does, without any artificial regularity assumption of the true solution. Numerical experiments are presented to demonstrate the utility of the method. (C) 2020 Elsevier B.V. All rights reserved.