Analysis and Efficient Implementation of Quadratic Spline Collocation ADI Methods for Variable-Order Time-Fractional Mobile-Immobile Diffusion Equations

Abstract

In this paper, a quadratic spline collocation (QSC) method combined with $L1$ time discretization in the framework of alternating direction implicit (ADI) approach, namely ADI-QSC-$L$1 method, is developed to solve the variable-order time-fractional mobile-immobile diffusion equations in multi-dimensional spaces. Discrete $L_2$ norm-based stability and error estimate are carefully discussed, which show that the proposed method is unconditionally stable and convergent with first-order accuracy in time and second-order accuracy in space. Then, based on the exponential-sum-approximation technique for the fast evaluation of the variable-order Caputo fractional derivative, an efficient implementation strategy of the ADI-QSC-$L1$ method, named ADI-QSC-${\rm F}L1$ is presented, which further improves the computational efficiency by reduced memory requirement and computational cost. Finally, numerical examples are provided to support both the theoretical results and efficiency of the developed method.