非线性波动方程的高效保能量数值算法

谢建强; 汪灿

doi:10.21656/1000-0887.460090

非线性波动方程的高效保能量数值算法

doi: 10.21656/1000-0887.460090

谢建强^,,
汪灿

安徽大学数学科学学院，合肥 230601

基金项目:

国家自然科学基金(12201005)

详细信息

作者简介:
谢建强(1988—)，男，副教授，博士，硕士生导师(通信作者. E-mail: xiejq1025@163.com).

通讯作者:
谢建强(1988—)，男，副教授，博士，硕士生导师(通信作者. E-mail: xiejq1025@163.com).

中图分类号: O241.82
计量
- 文章访问数: 16
- HTML全文浏览量: 4
- PDF下载量: 6
- 被引次数: 0
出版历程
- 收稿日期: 2025-05-06
- 修回日期: 2025-06-17
- 网络出版日期: 2026-01-21
- 刊出日期: 2026-01-01

An Efficient Energy-Preserving Numerical Algorithm for Nonlinear Wave Equations

XIE Jianqiang^,,
WANG Can

School of Mathematical Sciences, Anhui University, Hefei 230601, P.R.China

Funds:

The National Science Foundation of China(12201005)

摘要

摘要: 将降阶法、Lagrange乘子方法和紧致差分法相结合，对非线性波动方程建立一个时间二阶和空间四阶收敛精度的保能量数值算法，证明所提算法保持原始能量守恒性质，并给出相应算法的计算步骤.数值算例验证所提算法的正确性和有效性.
- 非线性波动方程 /
- 降阶法 /
- Lagrange乘子方法 /
- 紧致差分法 /
- 保能量数值算法
Abstract: An energy-preserving numerical algorithm, which is 2nd-order in time and 4th-order in space, for nonlinear wave equations was developed based on the order reduction method, the Lagrange multiplier method and the compact difference method. The discrete original energy conservation property of the suggested algorithm was proven. The computational procedure of the associated algorithm was exhibited. Numerical results validate the exactness and effectiveness of the proposed algorithm.
- nonlinear wave equation /
- order reduction method /
- Lagrange multiplier method /
- compact finite difference method /
- energy-preserving numerical algorithm

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参考文献(23)

[2]DRAZIN P J, JOHNSON R S. Solitons: An Introduction[M]. Cambridge: Cambridge University Press, 1989.

BRENNER P, VON WAHL W. Global classical solutions of nonlinear wave equations[J]. Mathematische Zeitschrift,1981,176(1): 87-121.

[3]WAZWAZ A M. New travelling wave solutions to the Boussinesq and the Klein-Gordon equations[J]. Communications in Nonlinear Science and Numerical Simulation,2008,13(5): 889-901.

[4]王媋瑗, 李宏, 何斯日古楞. 非线性sine-Gordon方程的连续时空混合有限元方法[J]. 应用数学和力学, 2024,45(4): 490-501. (WANG Chunyuan, LI Hong, HE Siriguleng. A continuous space-time mixed finite element method for sine-Gordon equations[J]. Applied Mathematics and Mechanics,2024,45(4): 490-501. (in Chinese))

[5]张宇, 邓子辰, 胡伟鹏. Sine-Gordon方程的多辛Leap-frog格式[J]. 应用数学和力学, 2013,34(5): 437-444.(ZHANG Yu, DENG Zichen, HU Weipeng. Multi-symplectic leap-frog scheme for sine-Gordon equation[J]. Applied Mathematics and Mechanics,2013,34(5): 437-444. (in Chinese))

[6]LI S, VU-QUOC L. Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation[J]. SIAM Journal on Numerical Analysis,1995,32(6): 1839-1875.

[7]MCLACHLAN R I, QUISPEL G R. Discrete gradient methods have an energy conservation law[J]. Discrete and Continuous Dynamical Systems,2014,34(3): 1099-1104.

[8]WANG B, WU X Y. The formulation and analysis of energy-preserving schemes for solving high dimensional nonlinear Klein-Gordon equations[J]. IMA Journal of Numerical Analysis,2019,39(4): 2016-2044.

[9]BRUGNANO L, FRASCA CACCIA G, IAVERNARO F. Energy conservation issues in the numerical solution of the semilinear wave equation[J]. Applied Mathematics and Computation,2015,270: 842-870.

[10]BRUGNANO L, ZHANG C J, LI D F. A class of energy-conserving Hamiltonian boundary value methods for nonlinear Schrdinger equation with wave operator[J]. Communications in Nonlinear Science and Numerical Simulation,2018,60: 33-49.

[11]QUISPEL G R W, MCLAREN D I. A new class of energy-preserving numerical integration methods[J]. Journal of Physics A: Mathematical General,2008,41(4): 045206.

[12]YANG X, ZHAO J, WANG Q. Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method[J]. Journal of Computational Physics,2017,333: 104-127.

[13]SHEN J, XU J, YANG J. The scalar auxiliary variable (SAV) approach for gradient flows[J]. Journal of Computational Physics,2018,353: 407-416.

[14]CAI W, JIANG C, WANG Y, et al. Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions[J]. Journal of Computational Physics,2019,395: 166-185.

[15]JIANG C, CAI W, WANG Y. A linearly implicit and local energy-preserving scheme for the sine-Gordon equation based on the invariant energy quadratization approach[J]. Journal of Scientific Computing,2019,80(3): 1629-1655.

[16]DENG D W, WANG Q H. A class of weighted energy-preserving Du Fort-Frankel difference schemes for solving sine-Gordon-type equations[J]. Communications in Nonlinear Science and Numerical Simulation,2023,117: 106916.

[17]DENG D W, CHEN J L, WANG Q H. Energy-preserving Du Fort-Frankel difference schemes for solving sine-Gordon equation and coupled sine-Gordon equations[J]. Numerical Algorithms,2023,93: 1045-1081.

[18]ZHANG X H, MEI L Q, GUO S M. Energy-conserving SAV-Hermite-Galerkin spectral scheme with time adaptive method for coupled nonlinear Klein-Gordon system in multi-dimensional unbounded domains[J]. Journal of Computational and Applied Mathematics,2025,454: 116204.

[19]CHENG Q, LIU C, SHEN J. A new Lagrange multiplier approach for gradient flows[J]. Computer Methods in Applied Mechanics and Engineering,2020,367: 113070.

[20]WANG Y U, JIN Y M, KHACHATURYAN A G. Phase field microelasticity modeling of dislocation dynamics near free surface and in heteroepitaxial thin films[J]. Acta Materialia,2003,51(14): 4209-4223.

[21]KARMA A, RAPPEL W J. Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics[J]. Physical Review E: Statistical Physics,Plasmas,Fluids and Related Interdisciplinary Topics,1996,53(4): R3017-R3020.

[22]BOETTINGER W J, WARREN J A, BECKERMANN C, KARMA A. Phase-field simulation of solidification[J]. Annual Review of Materials Research,2002,32: 163-194.

[23]孙志忠. 偏微分方程数值解法[M]. 第2版. 北京: 科学出版社, 2012.(SUN Zhizhong. Numerical Solution of Partial Differential Equation[M]. 2nd ed. Beijing: Science Press, 2012. (in Chinese))

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