Accelerated Spectral Deferred Correction Methods for Nonlinear Space Fractional Partial Differential Equations

In this paper, an efficient and accurate framework for nonlinear spacetime fractional diffusion equations is proposed. The methods are based on the spectral deferred correction technique, which employs a compact difference scheme as the preconditioner via the Picard integral collocation formulation. The nonlinear term is incorporated into the preconditioner in a way similar to linear systems without using Newtonian methods. The preconditioner is proven to be a stable operator, and the resulting spectral deferred correction method maintains an arbitrary order of accuracy and excellent stability. Due to the dense property of the central finite difference approximation of the fractional Laplacian (−Δ)s, a dual accelerated algorithm for the exact computation of the matrix–vector product is presented by introducing the discrete sine transform. The numerical results demonstrate that the proposed new methods are highly efficient and precise.