Alternating direction implicit-spectral element method (ADI-SEM) for solving multi-dimensional generalized modified anomalous sub-diffusion equation

Abstract

The main aim of the current paper is to solve the multi-dimensional generalized modified anomalous sub-diffusion equation by using a new spectral element method. At first, the time variable has been discretized by a finite difference scheme with second-order accuracy. The stability and convergence of the time-discrete scheme have been investigated. We show that the time-discrete scheme is unconditionally stable and the convergence order is O(τ2) in the temporal direction. Secondly, the Galerkin spectral element method has been combined with alternating direction implicit idea to discrete the space variable. The unconditional stability and convergence of the full-discrete scheme have been proved. By developing the proposed scheme, we need to calculate one-dimensional integrals for two-dimensional problems and two-dimensional integrals for three-dimensional problems. Thus, the used CPU time for the presented numerical procedure is lower than the two- and three-dimensional Galerkin spectral element methods. Also, the proposed method is suitable for computational domains obtained from the tensor product. Finally, two examples are analyzed to check the theoretical results.