1. An efficient numerical method on modified space-time sparse grid for time-fractional diffusion equation with nonsmooth data.
- Author
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Zhu BY, Xiao AG, and Li XY
- 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., Competing Interests: Competing interestsThe authors declare no competing interests., (© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023, Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.)
- Published
- 2023
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