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.

翻译：本文从理论和数值角度探讨了描述从次扩散（当 $t\rightarrow 0$ 时）到正常扩散（当 $t\rightarrow \infty$ 时）交叉行为的时间分数阶次扩散-正常输运方程。首先，利用双变量Mittag-Leffler函数研究了模型的适定性和正则性。理论结果表明，引入一阶导数算子后，解的正则性得到本质性提升。随后，针对初值光滑与非光滑情形，我们开发了高精度数值格式。具体而言，采用参数化双曲围道的围道积分法近似时间局部与非局部算子，并运用标准Galerkin有限元方法进行空间离散。严格误差分析表明，所提数值格式在时间方向达到谱精度，在空间方向达到最优收敛阶。此外，通过引入重心拉格朗日插值进一步优化算法并降低计算成本。最后，通过一维和二维数值实验验证了理论结果和加速算法的有效性，同时表明本文发展的数值格式具有高效性和鲁棒性。