Analyses of the Contour Integral Method for Time Fractional Normal-Subdiffusion Transport Equation

In this work, we theoretically and numerically discuss a class of time fractional normal-subdiffusion transport equation, which depicts a crossover from normal diffusion (as t -> 0) to sub-diffusion (as t -> infinity). 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 nomatter 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 spatial 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.

Automated summary

In this paper, we discuss a time fractional normal-subdiffusion transport equation and propose a high-precision numerical scheme. By introducing a first-order derivative operator and using a hyperbolic contour integral method, we improve the regularity of the solution and achieve good accuracy and convergence in both time and space for the numerical scheme. Numerical experiments validate the effectiveness and robustness of the theoretical results and algorithm.