An efficient numerical method on modified space-time sparse grid for time-fractional diffusion equation with nonsmooth data

Bi-Yun Zhu; Ai-Guo Xiao; Xue-Yang Li

doi:10.1007/s11075-023-01547-4

An efficient numerical method on modified space-time sparse grid for time-fractional diffusion equation with nonsmooth data

Numer Algorithms. 2023 May 11:1-36. doi: 10.1007/s11075-023-01547-4. Online ahead of print.

Authors

Bi-Yun Zhu¹, Ai-Guo Xiao¹, Xue-Yang Li¹

Affiliation

¹ School of Mathematics and Computational Science & National Center for Applied Mathematics in Hunan & Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan 411105 China.

Abstract

In this paper, we focus on developing a high efficient algorithm for solving d-dimension time-fractional diffusion equation (TFDE). For TFDE, the initial function or source term is usually not smooth, which can lead to the low regularity of exact solution. And such low regularity has a marked impact on the convergence rate of numerical method. In order to improve the convergence rate of the algorithm, we introduce the space-time sparse grid (STSG) method to solve TFDE. In our study, we employ the sine basis and the linear element basis for spatial discretization and temporal discretization, respectively. The sine basis can be divided into several levels, and the linear element basis can lead to the hierarchical basis. Then, the STSG can be constructed through a special tensor product of the spatial multilevel basis and the temporal hierarchical basis. Under certain conditions, the function approximation on standard STSG can achieve the accuracy order $O (2^{- J} J)$ with $O (2^{J} J)$ degrees of freedom (DOF) for $d = 1$ and $O (2^{Jd})$ DOF for $d > 1$ , where J denotes the maximal level of sine coefficients. However, if the solution changes very rapidly at the initial moment, the standard STSG method may reduce accuracy or even fail to converge. To overcome this, we integrate the full grid into the STSG, and obtain the modified STSG. Finally, we obtain the fully discrete scheme of STSG method for solving TFDE. The great advantage of the modified STSG method can be shown in the comparative numerical experiment.

Keywords: Hierarchical basis; L1 difference method; Sine pseudospectral method; Space-time sparse grid; Time-fractional diffusion equation.

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