非均匀网格上时间分数阶扩散—波动方程的 BDF2 型有限元方法

数学物理学报 ›› 2025, Vol. 45 ›› Issue (4): 1268-1290.

非均匀网格上时间分数阶扩散—波动方程的 BDF2 型有限元方法

祝鹏¹(),陈艳萍^2,^*(),徐先宇³()

¹嘉兴大学数据科学学院浙江嘉兴 314001
²南京邮电大学理学院南京 210023
³湘潭大学数学与计算科学学院湖南湘潭 411100

收稿日期:2024-09-19 修回日期:2025-01-30 出版日期:2025-08-26 发布日期:2025-08-01
通讯作者: *E-mail: ypchen@njupt.edu.cn
作者简介:E-mail: pzh@zjxu.edu.cn;|xuxianyuqdu@163.com
基金资助:
国家自然科学基金天元数学访问学者项目(12426616);浙江省自然科学基金(LY23A010005);南京邮电大学引进人才科研启动基金(NY223127)

BDF2-Type Finite Element Method for Time-Fractional Diffusion-Wave Equations on Nonuniform Grids

Zhu Peng¹(),Chen Yanping^2,^*(),Xu Xianyu³()

¹School of Data Science, Jiaxing University, Zhejiang Jiaxing 314001
²School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023
³School of Mathematics and Computational Science, Xiangtan University, Hunan Xiangtan 411100

Received:2024-09-19 Revised:2025-01-30 Online:2025-08-26 Published:2025-08-01
Supported by:
Visiting scholar program of National Natural Science Foundation of China(12426616);Zhejiang Provincial Natural Science Foundation of China(LY23A010005);Natural Science Research Start-up Foundation of Recruiting Talents of Nanjing University of Posts and Telecommunications(NY223127)

摘要/Abstract

摘要：

众所周知, 非均匀网格的研究可以有效地解决分数阶 Caputo 型导数的初值奇异现象. 在非均匀网格的理论分析中, 经常采用分数阶离散 Grönwall 不等式进行误差分析, 缺乏对误差结构的具体研究. 设计了一种非均匀网格上的误差卷积结构, 用于分析时间分数阶扩散-波动方程. 将二次插值近似应用于 Caputo 型导数, 通过使用降阶法和离散互补卷积核对 Caputo 型导数进行离散, 得到了非均匀网格上的 BDF2 型有限元方法. 离散互补卷积核在算法的收敛性分析中至关重要, 因为它简化有限元理论分析的过程, 并基于卷积核和插值估计的性质构建了全局一致性误差估计. 详细估计了非均匀网格上 BDF2 有限元格式的 $L^2$-范数误差和 $H^1$-范数误差, 并通过实验验证了所提出的有限元格式与理论收敛阶之间的一致性.

关键词: 时间分数阶扩散-波动方程, 离散卷积核, BDF2 型有限元格式, 误差卷积结构, 非均匀网格

Abstract:

As is well known, the study of nonuniform grids can effectively solve the initial value singularity phenomenon of fractional Caputo -type derivatives. In the theoretical analysis of nonuniform grids, fractional discrete Grönwall inequality is often used for error analysis, but there is a lack of specific research on error structures. An error convolution structure (ECS) was designed on nonuniform grids for analyzing the time fractional diffusion wave equation. A quadratic interpolation approximation was applied Caputo -type derivatives, and the BDF2 -type finite element method on nonuniform grids was obtained by discretizing it using a reduction method and a discrete complementary convolution kernel. The discrete complementary convolution kernel is crucial in the convergence analysis of algorithms, as it simplify the process of finite element theory analysis and construct global consistency errors based on the properties of convolution kernels and interpolation estimates. The $L^2$-norm error and $H^1$-norm error of the BDF2 finite element scheme on nonuniform grids were estimated in detail, and verifies the consistency between the proposed finite element scheme and the theoretical convergence order through experiments.

Key words: time fractional diffusion-wave equation, discrete complementary convolution kernels, BDF2 finite element format, error convolutional structure, nonuniform grids

中图分类号:

O241.8

祝鹏, 陈艳萍, 徐先宇. 非均匀网格上时间分数阶扩散—波动方程的 BDF2 型有限元方法[J]. 数学物理学报, 2025, 45(4): 1268-1290.

Zhu Peng, Chen Yanping, Xu Xianyu. BDF2-Type Finite Element Method for Time-Fractional Diffusion-Wave Equations on Nonuniform Grids[J]. Acta mathematica scientia,Series A, 2025, 45(4): 1268-1290.

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