Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation

doi:10.1007/s42967-022-00193-2

Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (4): 1509-1530.doi: 10.1007/s42967-022-00193-2

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Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation

Xiaoyi Liu, Tingchun Wang, Shilong Jin, Qiaoqiao Xu

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, Jiangsu, China

收稿日期:2021-11-11 修回日期:2022-02-26 出版日期:2022-12-20 发布日期:2022-09-26
通讯作者: Tingchun Wang,E-mail:wangtingchun2010@gmail.com E-mail:wangtingchun2010@gmail.com
基金资助:
This work is supported by the National Natural Science Foundation of China under Grant No. 11571181, and the Natural Science Foundation of Jiangsu Province of China under Grant No. BK20171454.

Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation

Xiaoyi Liu, Tingchun Wang, Shilong Jin, Qiaoqiao Xu

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, Jiangsu, China

Received:2021-11-11 Revised:2022-02-26 Online:2022-12-20 Published:2022-09-26
Supported by:
This work is supported by the National Natural Science Foundation of China under Grant No. 11571181, and the Natural Science Foundation of Jiangsu Province of China under Grant No. BK20171454.

摘要/Abstract

摘要： In this paper, two fourth-order compact finite difference schemes are derived to solve the nonlinear fourth-order wave equation which can be viewed as a generalized model from the nonlinear beam equation. Differing from the existing compact finite difference schemes which preserve the total energy in a recursive sense, the new schemes are proved to perfectly preserve the total energy in the discrete sense. By using the standard energy method and the cut-off function technique, the optimal error estimates of the numerical solutions are established, and the convergence rates are of O(h⁴ + τ²) with mesh-size h and time-step τ. In order to improve the computational efficiency, an iterative algorithm is proposed as the outer solver and the double sweep method for pentadiagonal linear algebraic equations is introduced as the inner solver to solve the nonlinear difference schemes at each time step. The convergence of the iterative algorithm is also rigorously analyzed. Several numerical results are carried out to test the error estimates and conservative properties.

关键词: Nonlinear fourth-order wave equation, Compact finite difference scheme, Error estimate, Energy conservation, Iterative algorithm

Abstract: In this paper, two fourth-order compact finite difference schemes are derived to solve the nonlinear fourth-order wave equation which can be viewed as a generalized model from the nonlinear beam equation. Differing from the existing compact finite difference schemes which preserve the total energy in a recursive sense, the new schemes are proved to perfectly preserve the total energy in the discrete sense. By using the standard energy method and the cut-off function technique, the optimal error estimates of the numerical solutions are established, and the convergence rates are of O(h⁴ + τ²) with mesh-size h and time-step τ. In order to improve the computational efficiency, an iterative algorithm is proposed as the outer solver and the double sweep method for pentadiagonal linear algebraic equations is introduced as the inner solver to solve the nonlinear difference schemes at each time step. The convergence of the iterative algorithm is also rigorously analyzed. Several numerical results are carried out to test the error estimates and conservative properties.

Key words: Nonlinear fourth-order wave equation, Compact finite difference scheme, Error estimate, Energy conservation, Iterative algorithm

中图分类号:

Xiaoyi Liu, Tingchun Wang, Shilong Jin, Qiaoqiao Xu. Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation[J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1509-1530.

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Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation

Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation

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