Communications on Applied Mathematics and Computation >

2021 , Vol. 3 >Issue 3: 429 - 444

DOI: https://doi.org/10.1007/s42967-020-00081-7

Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations

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  • 1 College of Science, Henan University of Technology, Zhengzhou 450001, China;
    2 School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, Fujian Province, China;
    3 College of Mathematics Sciences, Xinjiang Normal University, Urumqi 830054, China

Received date: 2020-03-06

  Revised date: 2020-05-13

  Online published: 2021-09-16

Supported by

This work was supported by the Fundamental Research Funds for the Henan Provincial Colleges and Universities in Henan University of Technology (2018RCJH10), the Training Plan of Young Backbone Teachers in Henan University of Technology (21420049), the Training Plan of Young Backbone Teachers in Colleges and Universities of Henan Province (2019GGJS094), the innovative Funds Plan of Henan University of Technology, Foundation of Henan Educational Committee (19A110005), and the National Natural Science Foundation of China (11861068).

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Abstract

The aim of this paper is to develop a fully discrete local discontinuous Galerkin method to solve a class of variable-order fractional diffusion problems. The scheme is discretized by a weighted-shifted Grünwald formula in the temporal discretization and a local discontinuous Galerkin method in the spatial direction. The stability and the L2-convergence of the scheme are proved for all variable-order α(t) ∈ (0, 1). The proposed method is of accuracy-order O(τ3 + hk+1), where τ, h, and k are the temporal step size, the spatial step size, and the degree of piecewise Pk polynomials, respectively. Some numerical tests are provided to illustrate the accuracy and the capability of the scheme.

Key words： Variable-order derivative; Discontinuous Galerkin method; Stability; Error estimates

Cite this article

Leilei Wei, Shuying Zhai, Xindong Zhang . Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations[J]. Communications on Applied Mathematics and Computation, 2021 , 3(3) : 429 -444 . DOI: 10.1007/s42967-020-00081-7

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