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

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 have 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 for spatial discretization, and all the sine coefficients can be divided into several levels. The sine coefficients with different levels are discretized by temporal basis with different scales, which can lead to the STSG method. Under certain conditions, the function approximation on standard STSG can achieve the accuracy order O(2-Jd) with O(2JJ) degrees of freedom (DOF) for d=1 and O(2Jd) DOF for d>1, where J denotes the maximal level of sine coefficients. However, the standard STSG is not suitable to simulate the singularity of TFDE at the initial time. To overcome this, we integrate the full grid into the STSG, and obtain the modified STSG. Then, the modified STSG is used to construct the fully discrete scheme for solving TFDE. The great advantage of STSG method can be shown in the comparative numerical experiment.Mathematics Subject Classification (2010) 35R11 · 65M70 · 65T40 · 68Q25