A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations

摘要

In this article, we apply the generalized BDF2-θ to the fractional mobile/immobile transport equations for its temporal discretization and the finite element method in the spatial direction. To derive the stability estimates and obtain the optimal error convergence rate, some properties of the convolution weights are proved based on which we show the scheme is unconditionally stable with an error of O(τ2+hr+1), where τ and h represent the temporal and spatial mesh size, respectively. We conduct exhaustive numerical tests to further confirm our theoretical analysis, and to overcome the initial singularity of the time fractional derivative we adopt the generalized BDF2-θ with starting parts in accordance with the framework of the shifted convolution quadrature (SCQ).