第三类Dirichlet边界下四阶抛物方程的紧差分格式

doi:10.6040/j.issn.1671-9352.0.2022.274

摘要/Abstract

摘要： 应用加权平均和高次Hermite插值等技术,提出逼近四阶导数的几个有用的数值微分公式,并对其截断误差进行分析。在此基础上建立求解第三类Dirichlet边界条件下四阶抛物方程初边值问题的三个高阶紧差分格式,应用Fourier分析方法证明格式的无条件稳定性,并对其进行数值验证。这三个差分格式的差异主要体现在空间导数临近边界处的离散方式不同,所得格式全局精度均达到了时间二阶、空间四阶。

关键词: 四阶抛物方程, 第三类Dirichlet边界, 高精度, 紧差分格式, 稳定性

Abstract: Based on some techniques involving the weighted average and high order Hermite interpolation, several useful differentiation formulae for approximating the fourth-order derivatives are derived along with the truncation error analyses. Then three high order compact difference schemes are proposed to solve the initial-boundary value problem of the fourth-order parabolic equations with the third Dirichlet boundary conditions. The unconditional stability is proved by the Fourier analysis method. Numerical experiments are carried out. The major difference of the proposed three schemes lies in the different numerical treatment of spatial derivatives near the boundary. The global accuracy of all presented schemes can attain the order of two in time and four in space.

Key words: fourth-order parabolic equation, third Dirichlet boundary condition, high accuracy, compact difference scheme, stability

中图分类号:

O241.82

黄钰,高广花. 第三类Dirichlet边界下四阶抛物方程的紧差分格式[J]. 《山东大学学报(理学版)》, 2023, 58(4): 16-28.

HUANG Yu, GAO Guang-hua. Compact difference schemes for the fourth-order parabolic equations with the third Dirichlet boundary[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(4): 16-28.

参考文献

[1] 张迪, 杨青. 一类四阶非线性抛物方程的紧致差分方法[J]. 山东师范大学学报(自然科学版), 2020, 35(2):190-196. ZHANG Di, YANG Qing. A compact difference scheme for a class of fourth order nonlinear parabolic equations[J]. Journal of Shandong Normal University(Natural Science), 2020, 35(2):190-196.
[2] HU Xiuling, ZHANG Luming. A new implicit compact difference scheme for the fourth-order fractional diffusion-wave system[J]. International Journal of Computer Mathematics, 2014, 91(10):2215-2231.
[3] GAO Guanghua, XU Peng, TANG Rui. Fast compact difference scheme for the fourth-order time multi-term fractional sub-diffusion equations with the first Dirichlet boundary[J]. Journal of Applied Analysis and Computation, 2021, 11(6):2736-2761.
[4] YAO Zhongsheng, WANG Zhibo. A compact difference scheme for fourth-order fractional sub-diffusion equations with Neumann boundary conditions[J]. Journal of Applied Analysis and Computation, 2018, 8(4):1159-1169.
[5] 陆宣如. 四阶抛物方程不同边界条件下定解问题的差分方法[D]. 南京:东南大学, 2021. LU Xuanru. The difference schemes for the fourth order parabolic equation with different boundary conditions[D]. Nanjing: Southeast University, 2021.
[6] VONG S, WANG Z B. Compact finite difference scheme for the fourth-order fractional subdiffusion system[J]. Advances in Applied Mathematics and Mechanics, 2014, 6(4):419-435.
[7] ARSHAD S, WALI M, HUANG J F, et al. Numerical framework for the Caputo time-fractional diffusion equation with fourth order derivative in space[J]. Journal of Applied Mathematics and Computing, 2022, 68:3295-3316.
[8] MOHANTY R K, KAUR D. High accuracy two-level implicit compact difference scheme for 1D unsteady biharmonic problem of first kind: application to the generalized Kuramoto-Sivashinsky equation[J]. Journal of Difference Equations and Applications, 2019, 25(2):243-261.
[9] JI Cuicui, SUN Zhizhong, HAO Zhaopeng. Numerical algorithms with high spatial accuracy for the fourth-order fractional sub-diffusion equations with the first Dirichlet boundary conditions[J]. Journal of Scientific Computing, 2016, 66(3):1148-1174.
[10] GAO Guanghua, TANG Rui, YANG Qian. A compact finite difference scheme for the fourth-order time multi-term fractional sub-diffusion equations with the first Dirichlet boundary conditions[J]. International Journal of Numerical Analysis and Modeling, 2021, 18(1):100-119.
[11] 单双荣. 解四阶抛物型方程的高精度差分格式[J]. 华侨大学学报(自然科学版), 2005, 26(1):19-22. SHAN Shuangrong. A family of high accurate difference scheme for solving four-order parabolic equation[J]. Journal of Huaqiao University(Natural Science), 2005, 26(1):19-22.
[12] 张星, 单双荣. 解四阶抛物型方程的高精度显式差分格式[J]. 华侨大学学报(自然科学版), 2010, 31(6):703-705. ZHANG Xing, SHAN Shuangrong. Explicit difference scheme of high accuracy for solving four-order parabolic equation[J]. Journal of Huaqiao University(Natural Science), 2010, 31(6):703-705.
[13] HU Xiuling, ZHANG Luming. On finite difference methods for fourth-order fractional diffusion-wave and subdiffusion systems[J]. Applied Mathematics and Computation, 2012, 218(9):5019-5034.
[14] MOHANTY R K, KAUR D, SINGH S. A class of two- and three-level implicit methods of order two in time and four in space based on half-step discretization for two-dimensional fourth order quasi-linear parabolic equations[J]. Applied Mathematics and Computation, 2019, 352:68-87.
[15] NANDAL S, PANDEY D N. Second order compact difference scheme for time fractional sub-diffusion fourth-order neutral delay differential equations[J]. Differential Equations and Dynamical Systems, 2021, 29(1):69-86.
[16] 孙志忠. 偏微分方程数值解法[M]. 3版. 北京:科学出版社,2022: 7-9. SUN Zhizhong. Numerical solutions of partial differential equations[M]. 3rd ed. Beijing: Science Press, 2022: 7-9.

相关文章 15

[1]	李晓伟,李桂花. 考虑环境病毒影响的COVID-19模型的动力学性态研究[J]. 《山东大学学报(理学版)》, 2023, 58(1): 10-15.
[2]	霍林杰,张存华. 具有Holling-Ⅲ型功能反应的捕食扩散系统的稳定性和Hopf分支[J]. 《山东大学学报(理学版)》, 2023, 58(1): 16-24.
[3]	孙春杰,张存华. 一类Beddington-DeAngelis-Tanner型扩散捕食系统的稳定性和Turing不稳定性[J]. 《山东大学学报(理学版)》, 2022, 57(9): 83-90.
[4]	苏晓艳,陈京荣,尹会玲. 广义区间值Pythagorean三角模糊集成算子及其决策应用[J]. 《山东大学学报(理学版)》, 2022, 57(8): 77-87.
[5]	庞玉婷,赵东霞,鲍芳霞. 具有多时滞和多参数的双向环状网络的稳定性[J]. 《山东大学学报(理学版)》, 2022, 57(8): 103-110.
[6]	韩卓茹,李善兵. 具有空间异质和合作捕食的捕食-食饵模型的正解[J]. 《山东大学学报(理学版)》, 2022, 57(7): 35-42.
[7]	孟旭东. 集合优化问题解集的稳定性和扩展适定性[J]. 《山东大学学报(理学版)》, 2022, 57(2): 98-110.
[8]	焦战,靳祯. 空间异质的非局部扩散SI传染病模型的动力学[J]. 《山东大学学报(理学版)》, 2022, 57(11): 70-77.
[9]	张钰倩,张太雷. 具有复发效应的SEAIR模型及在新冠肺炎传染病中的应用[J]. 《山东大学学报(理学版)》, 2022, 57(1): 56-68.
[10]	沈维,张存华. 时滞食饵-捕食系统的多次稳定性切换和Hopf分支[J]. 《山东大学学报(理学版)》, 2022, 57(1): 42-49.
[11]	朱彦兰,周伟,褚童,李文娜. 管理委托下的双寡头博弈的复杂动力学分析[J]. 《山东大学学报(理学版)》, 2021, 56(7): 32-45.
[12]	周艳,张存华. 具有集群行为的捕食者-食饵反应扩散系统的稳定性和Turing不稳定性[J]. 《山东大学学报(理学版)》, 2021, 56(7): 73-81.
[13]	郭慧瑛,杨富霞,张翠萍. 有有限Ext-强Ding投射维数的模的稳定性[J]. 《山东大学学报(理学版)》, 2021, 56(4): 31-38.
[14]	唐洁,魏玲,任睿思,赵思雨. 基于可能属性分析的粒描述[J]. 《山东大学学报(理学版)》, 2021, 56(1): 75-82.
[15]	阳忠亮, 郭改慧. 一类带有B-D功能反应的捕食-食饵模型的分支分析[J]. 《山东大学学报(理学版)》, 2020, 55(7): 9-15.

多维度评价

Viewed

Full text

Abstract

Cited

Shared

Discussed