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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2023, Vol. 40 ›› Issue (5): 793-806.doi: 10.3969/j.issn.1005-3085.2023.05.008

• • 上一篇    下一篇

缓增分数阶扩散方程的高阶时间离散LDG方法

李敏敏1,  李  灿1,  赵丽静2   

  1. 1. 西安理工大学理学院，西安 710054;
    2. 西北工业大学数学与统计学院，西安 710129
  • 收稿日期:2021-04-15 接受日期:2022-09-30 出版日期:2023-10-15 发布日期:2023-12-15
  • 通讯作者: 李 灿 E-mail: mathlican@xaut.edu.cn
  • 基金资助:
    国家自然科学基金(11801148)；陕西省自然科学基金(2023-JC-YB-045).

LDG Method with High Order Time Stepping Scheme for a Time Tempered Fractional Diffusion Equation

LI Minmin1,  LI Can1,  ZHAO Lijing2   

  1. 1. School of Science, Xi'an University of Technology, Xi'an 710054;
    2. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710129
  • Received:2021-04-15 Accepted:2022-09-30 Online:2023-10-15 Published:2023-12-15
  • Contact: C. Li. E-mail address: mathlican@xaut.edu.cn
  • Supported by:
    The National Natural Science Foundation of China (11801148); the Natural Science Basic Research Plan in Shaanxi Province (2023-JC-YB-045).

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摘要：

研究了缓增分数阶扩散方程的高阶时间离散局部间断Galerkin(Local Discontinuous Galerkin, LDG)方法，不是直接求解缓增分数阶扩散方程，而是首先通过变换将其转化成Caputo型时间分数阶扩散方程。接着，采用L1-2差分逼近离散Caputo型分数阶导数，间断有限元离散空间变量，构造求解模型的全离散LDG格式。证明了所建立的全离散格式为无条件稳定的且具有最优误差阶，两个数值算了验证了所建立数值格式的精度和鲁棒性。数值实验结果表明所建立格式在时间和空间方向均具有高精度。

关键词: 局部间断有限元方法, 缓增分数阶扩散方程, 稳定性, 收敛性

Abstract:

In the present paper, we develop a local discontinuous Galerkin (LDG) method  with a high order time stepping scheme for a time tempered fractional diffusion equation. Instead of solving the present model directly, we first transform it into a diffusion equation with Caputo fractional derivative. Then, the full-discrete LDG is constructed by using the L1-2 time stepping scheme to approach the Caputo fractional derivative, and using the discontinuous Galerkin to approximate the space derivative. We prove that the full-discrete discontinuous Galerkin method is unconditionally stable with the optimal convergence rate. We present two numerical examples to illustrate the accuracy and the robustness of the numerical method proposed in this paper. Our experimental results show that the high order accuracy of the present numerical scheme are obtained in both time and space variables.

Key words: local discontinuous Galerkin methods, tempered fractional diffusion equation, stability, convergence

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