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High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Diffusion Equation

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Abstract

In this article, some high-order local discontinuous Galerkin (LDG) schemes based on some second-order \( \theta \) approximation formulas in time are presented to solve a two-dimensional nonlinear fractional diffusion equation. The unconditional stability of the LDG scheme is proved, and an a priori error estimate with \(O(h^{k+1}+\varDelta t^2)\) is derived, where \(k\geqslant 0\) denotes the index of the basis function. Extensive numerical results with \(Q^k(k=0,1,2,3)\) elements are provided to confirm our theoretical results, which also show that the second-order convergence rate in time is not impacted by the changed parameter \(\theta\).

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Acknowledgements

The authors are grateful to the two anonymous referees and editor for their valuable comments which greatly improved the presentation of the paper. This work is supported by the National Natural Science Foundation of China (11661058, 11761053), the Natural Science Foundation of Inner Mongolia (2017MS0107), and the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (NJYT-17-A07).

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian Province, China

    Min Zhang

  2. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, China

    Yang Liu & Hong Li

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  1. Min Zhang
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  2. Yang Liu
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Correspondence to Yang Liu.

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Zhang, M., Liu, Y. & Li, H. High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Diffusion Equation. Commun. Appl. Math. Comput. 2, 613–640 (2020). https://doi.org/10.1007/s42967-019-00058-1

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