JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (4): 16-28.doi: 10.6040/j.issn.1671-9352.0.2022.274

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Compact difference schemes for the fourth-order parabolic equations with the third Dirichlet boundary

HUANG Yu, GAO Guang-hua*   

  1. School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, Jiangsu, China
  • Published:2023-03-27

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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

CLC Number: 

  • O241.82
[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.
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