Analyses of the contour integral method for time fractional subdiffusion-normal transport equation

TL;DR

This paper presents a high-precision numerical scheme based on contour integral methods for solving the time fractional subdiffusion-normal transport equation, demonstrating spectral accuracy and efficiency through theoretical analysis and numerical experiments.

Contribution

The work introduces a novel high-precision contour integral method combined with finite element discretization for the fractional diffusion equation, improving accuracy and reducing computational cost.

Findings

Spectral accuracy in time achieved by the proposed scheme

Optimal convergence order in space demonstrated

Algorithm effectively accelerates computations with verified robustness

Abstract

In this work, we theoretically and numerically discuss the time fractional subdiffusion-normal transport equation, which depicts a crossover from sub-diffusion (as $t\rightarrow 0$) to normal diffusion (as $t\rightarrow \infty$). Firstly, the well-posedness and regularities of the model are studied by using the bivariate Mittag-Leffler function. Theoretical results show that after introducing the first-order derivative operator, the regularity of the solution can be improved in substance. Then, a numerical scheme with high-precision is developed no matter the initial value is smooth or non-smooth. More specifically, we use the contour integral method (CIM) with parameterized hyperbolic contour to approximate the temporal local and non-local operators, and employ the standard Galerkin finite element method for spacial discretization. Rigorous error estimates show that the proposed numerical scheme has spectral accuracy in time and optimal convergence order in space. Besides, we further improve the algorithm and reduce the computational cost by using the barycentric Lagrange interpolation. Finally, the obtained theoretical results as well as the acceleration algorithm are verified by several 1-D and 2-D numerical experiments, which also show that the numerical scheme developed in this paper is effective and robust.