阻尼eKdV-Burgers方程的共形广义多辛Fourier拟谱算法

李骞; 王桂霞; 王一辰

doi:10.21656/1000-0887.450263

阻尼eKdV-Burgers方程的共形广义多辛Fourier拟谱算法

doi: 10.21656/1000-0887.450263

李骞^1,,
王桂霞^{1, 2, 3, ,},
王一辰¹

1.
内蒙古师范大学数学科学学院, 呼和浩特 010022
2.
内蒙古自治区应用数学中心, 呼和浩特 010022
3.
无穷维哈密顿系统及其算法应用教育部重点实验室, 呼和浩特 010022

基金项目:

国家自然科学基金 62161045

内蒙古自治区自然科学基金重点项目 2022ZD05

内蒙古自治区自然科学基金 2023LHMS01007

内蒙古自治区自然科学基金 2022JBQN074

详细信息

作者简介:
李骞(2000—), 女, 硕士生(E-mail: 3491240780@qq.com)

通讯作者:
王桂霞(1968—), 女, 教授, 博士(通讯作者. E-mail: nsdwgx@126.com)

中图分类号: O241.82
计量
- 文章访问数: 77
- HTML全文浏览量: 16
- PDF下载量: 9
- 被引次数: 0
出版历程
- 收稿日期: 2024-09-29
- 修回日期: 2024-12-24
- 刊出日期: 2025-11-01

A Conformal Generalized Multi-Symplectic Fourier Pseudo-Spectral Algorithm for Damping eKdV-Burgers Equations

LI Qian^1
,,
WANG Guixia^{1, 2, 3
, ,},
WANG Yichen¹

1.
College of Mathematics Science, Inner Mongolia Normal University, Hohhot 010022, P.R.China
2.
Center for Applied Mathematics Science, Inner Mongolia, Hohhot 010022, P.R.China
3.
Laboratory of Infinite-Dimensional Hamiltonian System and Its Algorithm Application, Ministry of Education, Hohhot 010022, P.R.China

摘要

摘要: 基于Hamilton共形广义多辛理论, 研究一类阻尼eKdV-Burgers方程的共形广义多辛Fourier拟谱格式的保结构算法. 首先, 通过引入中间变量, 将方程转化为满足局部守恒的共形广义多辛Hamilton系统, 并利用Strang分裂方法, 将其分裂为守恒子系统和耗散子系统. 进一步, 空间上利用Fourier拟谱方法, 时间上利用隐中点方法, 对该系统进行离散，得到共形广义多辛Fourier拟谱格式, 在周期边界条件下, 该格式满足全局共形质量守恒律和动量守恒律. 数值实例表明该算法是有效的, 能够保持系统质量和动量衰减特性.
- 耗散项 /
- 浅水效应 /
- eKdV-Burgers方程 /
- 共形广义多辛 /
- Fourier拟谱方法 /
- Strang分裂
Abstract: Based on the conformal generalized multi-symplectic theory for Hamiltonian systems, a class of conformal generalized multi-symplectic pseudo-spectral algorithms for damping eKdV-Burgers the equations were studied. Firstly, through introduction of intermediate variables, the equation was transformed into a conformal generalized multi-symplectic Hamiltonian system satisfying local conservation, and the Strang splitting method was used to split it into a conservative subsystem and a dissipative subsystem. Furthermore, the Fourier pseudo-spectral method was applied spatially and the hidden midpoint method applied temporally to discretize the system to obtain the conformal generalized multi-symplectic Fourier pseudo-spectral scheme, which meets the global conformal mass conservation law and the momentum conservation law under the periodic boundary conditions. Numerical examples show that, the algorithm is effective and can maintain the mass and momentum decay characteristics of the system.
- dissipative term /
- shallow water effect /
- eKdV-Burgers equation /
- conformal generalized multi-symplectic /
- Fourier pseudo-spectral method /
- Strang split

HTML全文

图 1 单孤立波的演化过程(算例1)

Figure 1. The evolution of a single solitary wave (case 1)

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图 2 不同时刻的波形图(算例1)

Figure 2. Waveform plots at different moments (case 1)

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图 3 全局共形质量随时间的变化(算例1)

Figure 3. The global conformal change mass over time (case 1)

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图 4 质量衰减速度误差(算例1)

Figure 4. The mass decay rate error (case 1)

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图 5 广义多辛误差(算例1)

Figure 5. Generalized multi-symplectic errors (case 1)

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图 6 局部能量误差(算例1)

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 6. Local energy errors (case 1)

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图 7 两种差分方法质量衰减速度r的误差对比

Figure 7. Comparison of the errors of mass decay rate r of the 2 difference methods

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图 8 τ=10^-4时的空间谱收敛精度测试

Figure 8. The spatial spectral convergence accuracy test at τ=10^-4

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图 9 单孤立波的演化过程(算例2)

Figure 9. The evolution of the single solitary wave (case 2)

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图 10 质量衰减速度的误差(算例2)

Figure 10. The error of the mass decay rate (case 2)

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图 11 局部能量误差(算例2)

Figure 11. Local energy errors (case 2)

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图 12 广义多辛误差(算例2)

Figure 12. Generalized multi-symplectic errors (case 2)

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图 13 非线性项的影响

Figure 13. The effects of nonlinear terms

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图 14 高阶非线性项的影响

Figure 14. The effects of higher-order nonlinear terms

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图 15 频散项的影响

Figure 15. The effects of the dispersion terms

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图 16 整体演化图

Figure 16. Overall evolution curves

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图 17 单孤立波的演化过程1

Figure 17. Evolution 1 of the single solitary wave

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图 18 不同时刻的波形图1

Figure 18. Waveform plots 1 at different moments

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图 19 单孤立波的演化过程2

Figure 19. The evolution 2 of the single solitary wave

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图 20 不同时刻的波形图2

Figure 20. Waveform plots 2 at different moments

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图 21 双孤立波的演化过程(算例3)

Figure 21. The evolution of double solitary waves (case 3)

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图 22 不同时刻的波形(算例3)

Figure 22. Wave forms of different moments (case 3)

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图 23 全局共形质量随时间变化(算例3)

Figure 23. Global conformal mass changes over time (case 3)

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图 24 质量衰减速度的误差(算例3)

Figure 24. The mass decay rate error (case 3)

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图 25 广义多辛误差图(算例3)

Figure 25. Generalized multi-symplectic errors (case 3)

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图 26 局部能量误差图(算例3)

Figure 26. Local energy errors (case 3)

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图 27 双孤立波的演化过程

Figure 27. The evolution of double solitary waves

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图 28 不同时刻波形对比

Figure 28. Comparison of waveforms at different moments

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图 29 双孤立波不同时刻(t=0, t=4.5)

Figure 29. The double solitary waves of different moments (t=0, t=4.5)

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图 30 双孤立波不同时刻(t=4.7, t=10)

Figure 30. The double solitary waves of different moments (t=4.7, t=10)

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表 1 取不同时间步长的误差和误差阶

Table 1. Errors and error orders at different time steps

h	τ	error	error order
1/4	1/16	3×10^-3	-
1/4	1/32	7.18×10^-4	1.958
1/4	1/64	1.93×10^-4	1.895
1/4	1/128	4.85×10^-5	1.994

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参考文献(39)

[1]	冯康, 秦孟兆. 哈密尔顿系统的辛几何算法[M]. 杭州: 浙江科学技术出版社, 2003. FENG Kang, QIN Mengzhao. Symplectic Geometric Algorithms for Hamiltonian Systems[M]. Hangzhou: Zhejiang Science and Technology Publishing House, 2003. (in Chinese)
[2]	秦孟兆, 王雨顺. 偏微分方程中的保结构算法[M]. 杭州: 浙江科学技术出版社, 2011. QIN Mengzhao, WANG Yushun. Structure-Preserving Algorithm for Partial Differential Equation[M]. Hangzhou: Zhejiang Science and Technology Publishing House, 2011. (in Chinese)
[3]	HAIRER E, LUBICH C, WANNER G. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations[M]. 2nd ed. Berlin Heidlberg: Springer, 2006.
[4]	MCLACHLAN R I, QUISPEL G R W. Splittingmethods[J]. Acta Numerica, 2002, 11 : 341-434. doi: 10.1017/S0962492902000053
[5]	王雨顺, 秦孟兆. 变分与无限维系统的高精度辛格式[J]. 计算数学, 2002, 24 (4): 431-436. WANG Yushun, QIN Mengzhao. Variation and high order accuracy symplectic scheme for infinite dimensional system[J]. Mathematica Numerica Sinica, 2002, 24 (4): 431-436. (in Chinese)
[6]	TANG Y F, VÁZQUEZ L, ZHANG F, et al. Symplectic methods for the nonlinear Schrödinger equation[J]. Computers & Mathematics With Applications, 1996, 32 (5): 73-83.
[7]	SHANG Z J. KAM theorem of symplectic algorithms for Hamiltoniansystems[J]. Numerische Mathematik, 1999, 83 (3): 477-496. doi: 10.1007/s002110050460
[8]	BRIDGES T J, REICH S. Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conservesymplecticity[J]. Physics Letters A, 2001, 284 (4/5): 184-193.
[9]	BRIDGES T J. Multi-symplectic structures and wavepropagation[J]. Mathematical Proceedings of the Cambridge Philosophical Society, 1997, 121 (1): 147-190. doi: 10.1017/S0305004196001429
[10]	MARSDEN J E, PATRICK G W, SHKOLLER S. Multisymplectic geometry, variational integrators, and nonlinear PDEs[J]. Communications in Mathematical Physics, 1998, 199 (2): 351-395. doi: 10.1007/s002200050505
[11]	MARSDEN J E, PEKARSKY S, SHKOLLER S, et al. Variational methods, multisymplectic geometry and continuum mechanics[J]. Journal of Geometry and Physics, 2001, 38 (3/4): 253-284.
[12]	王雨顺, 王斌, 秦孟兆. 2+1维sine-Gordon方程多辛格式的复合构造[J]. 中国科学(A辑), 2003, 33 (3): 272-281. WANG Yushun, WANG Bin, QIN Mengzhao. Composite construction of multi-symplectic scheme for 2+1 dimensional sine-Gordon equation[J]. Science in China (Serises A), 2003, 33 (3): 272-281. (in Chinese)
[13]	王雨顺, 王斌, 季仲贞. 孤立波方程的保结构算法[J]. 计算物理, 2004, 21 (5): 386-400. WANG Yushun, WANG Bin, JI Zhongzhen. Structure preserving algorithms for soliton equations[J]. Chinese Journal of Computation Physics, 2004, 21 (5): 386-400. (in Chinese)
[14]	王雨顺, 王斌, 秦孟兆. 偏微分方程的局部保结构算法[J]. 中国科学(A辑): 数学, 2008, 38 (4): 377-397. WANG Yushun, WANG Bin, QIN Mengzhao. Local structure-preserving algorithm for partial differential equations[J]. Science in China (Series A): Mathematics, 2008, 38 (4): 377-397. (in Chinese)
[15]	SUN Y J, QIN M Z. A multi-symplectic scheme for RLW equation[J]. Journal of Computational Mathematics, 2004, 22 (4): 611-621.
[16]	胡伟鹏, 邓子辰. 无限维动力学系统的保结构分析方法[M]. 西安: 西北工业大学出版社, 2015. HU Weipeng, DENG Zichen. Structure-Preserving Analysis Method for Infinite Dimensional Dynamic Systems[M]. Xi'an: Northwestern Polytechnical University Press, 2015. (in Chinese)
[17]	胡伟鹏, 邓子辰, 李文成. KdV方程的多辛算法及其孤子解的数值模拟[J]. 西北工业大学学报, 2008, 26 (1): 128-131. HU Weipeng, DENG Zichen, LI Wencheng. Multi-symplectic algorithm and simulation of soliton for KdV equation[J]. Journal of Northwestern Polytechnical University, 2008, 26 (1): 128-131. (in Chinese)
[18]	何啸, 贾村, 孟静, 等. 内孤立波过陆架陆坡地形的数值模拟研究[J]. 海洋科学, 2023, 47 (3): 1-14. HE Xiao, JIA Cun, MENG Jing, et al. Numerical investigation of the evolution of internal solitary waves over shelf-slope topography[J]. Marine Sciences, 2023, 47 (3): 1-14. (in Chinese)
[19]	郭峰, 吴凤珍. MKdV方程的多辛格式[J]. 河南师范大学学报(自然科学版), 2005, 33 (1): 128-129. GUO Feng, WU Fengzhen. Multi-symplectic scheme for MKdV equation[J]. Journal of Henan Normal University (Natural Science Edition), 2005, 33 (1): 128-129. (in Chinese)
[20]	胡伟鹏, 邓子辰, 李文成. 广义KdV-mKdV方程的多辛算法及孤波解数值模拟[J]. 西北工业大学学报, 2008, 26 (4): 450-453. HU Weipeng, DENG Zichen, LI Wencheng. Multi-symplectic scheme and simulation of solitary wave solution for generalized KdV-mKdV equation[J]. Journal of Northwestern Polytechnical University, 2008, 26 (4): 450-453. (in Chinese)
[21]	王晨逦, 王桂霞, 萨和雅. EKdV方程的高阶多辛保结构算法及孤立波解的数值模拟[J]. 内蒙古师范大学学报(自然科学汉文版), 2023, 52 (2): 119-126. WANG Chenli, WANG Guixia, SA Heya. High order multisymplectic structure-preserving algorithm and numerical simulation of solitary wave solutions for EKdV equation[J]. Journal of Inner Mongolia Normal University (Natural Science Edition), 2023, 52 (2): 119-126. (in Chinese)
[22]	HELFRICH K R, MELVILLE W K. On long nonlinear internal waves over slope-shelf topography[J]. Journal of Fluid Mechanics, 1986, 167 : 285. doi: 10.1017/S0022112086002823
[23]	MCLACHLAN R, PERLMUTTER M. Conformal Hamiltonian systems[J]. Journal of Geometry and Physics, 2001, 39 (4): 276-300. doi: 10.1016/S0393-0440(01)00020-1
[24]	BHATT A, FLOYD D, MOORE B E. Second order conformal symplectic schemes for damped Hamiltonian systems[J]. Journal of Scientific Computing, 2016, 66 (3): 1234-1259. doi: 10.1007/s10915-015-0062-z
[25]	MOORE B E. Conformal multi-symplectic integration methods for forced-damped semi-linear wave equations[J]. Mathematics and Computers in Simulation, 2009, 80 (1): 20-28. doi: 10.1016/j.matcom.2009.06.024
[26]	MOORE B E, NOREÑA L, SCHOBER C M. Conformal conservation laws and geometric integration for damped Hamiltonian PDEs[J]. Journal of Computational Physics, 2013, 232 (1): 214-233. doi: 10.1016/j.jcp.2012.08.010
[27]	MOORE B E. Multi-conformal-symplectic PDEs and discretizations[J]. Journal of Computational and Applied Mathematics, 2017, 323 : 1-15. doi: 10.1016/j.cam.2017.04.008
[28]	汪佳玲. 几个偏微分方程的保结构算法构造及误差分析[D]. 南京: 南京师范大学, 2017. WANG Jialing. Construction and analysis of the structure-preserving algorithms for the Hamiltonian partial differential equations[D]. Nanjing: Nanjing Normal University, 2017. (in Chinese)
[29]	郑家栋, 张汝芬, 郭本瑜. SRLW方程的Fourier拟谱方法[J]. 应用数学和力学, 1989, 10 (9): 801-810. ZHENG Jiadong, ZHANG Rufen, GUO Benyu. The Fourier pseudo-spectral method for the SRLW equation[J]. Applied Mathematics and Mechanics, 1989, 10 (9): 801-810. (in Chinese)
[30]	邓镇国, 马和平. 广义KdV方程Fourier谱逼近的最优误差估计[J]. 应用数学和力学, 2009, 30 (1): 30-39. DENG Zhenguo, MA Heping. Optimal error estimates for Fourier spectral approxiation of the generalized KdV equation[J]. Applied Mathematics and Mechanics, 2009, 30 (1): 30-39. (in Chinese)
[31]	HU D D, CAI W J, XU Z Z, et al. Dissipation-preserving Fourier pseudo-spectral method for the space fractional nonlinear sine-Gordon equation with damping[J]. Mathematics and Computers in Simulation, 2021, 188 : 35-59. doi: 10.1016/j.matcom.2021.03.034
[32]	HE Y, ZHAO X F. Numerical methods for some nonlinear Schrödinger equations in soliton management[J]. Journal of Scientific Computing, 2023, 95 (2): 61. doi: 10.1007/s10915-023-02181-x
[33]	许壮志. 空间分数阶非线性Schrödinger方程保结构算法[D]. 南京: 南京师范大学, 2021. XU Zhuangzhi. Structure-preserving algorithms for the space fractional nonlinear Schrödinger equation[D]. Nanjing: Nanjing Normal University, 2021. (in Chinese)
[34]	蔡树群. 内孤立波数值模式及其在南海区域的应用[M]. 北京: 海洋出版社, 2015. CAI Shuqun. Numerical Model of Internal Solitary Waves and Its Application in South China Sea Region[M]. Beijing: Ocean Press, 2015. (in Chinese)
[35]	方欣华, 杜涛. 海洋内波基础和中国海内波[M]. 青岛: 中国海洋大学出版社, 2005. FANG Xinhua, DU Tao. Fundamentals of Oceanic Internal Waves and Internal Waves in the China Seas[M]. Qingdao: China Ocean University Press, 2005. (in Chinese)
[36]	张善武. 基于变系数KdV-type理论模型的南海北部内孤立波传播演变过程研究[D]. 青岛: 中国海洋大学, 2014. ZHANG Shanwu. Investigation of the propagation and evolution processes of internal solitary waves in the Northern South China Sea based on the variable-coefficients KdV-type theoretical models[D]. Qingdao: Ocean University of China, 2014. (in Chinese)
[37]	傅浩. 共形Hamilton系统的若干保结构算法研究[D]. 长沙: 国防科技大学, 2018. FU Hao. Some structure-preserving algorithms for conformal Hamiltonian system[D]. Changsha: National University of Defense Technology, 2018. (in Chinese)
[38]	宋松和, 陈亚铭, 朱华君. KdV方程的多辛Fourier拟谱格式及其孤立波解的数值模拟[J]. 安徽大学学报(自然科学版), 2010, 34 (4): 1-7. SONG Songhe, CHEN Yaming, ZHU Huajun. Multi-symplectic Fourier pseudospectral scheme and simulation of soliton solutions for the KdV equation[J]. Journal of Anhui University (Natural Sciences Edition), 2010, 34 (4): 1-7. (in Chinese)
[39]	向新民. 谱方法的数值分析[M]. 北京: 科学出版社, 2000. XIANG Xinmin. Numerical Analysis of Spectral Method[M]. Beijing: Science Press, 2000. (in Chinese)