A New Finite Difference Method for Rosenau-Burgers Equation

Abstract

Abstract: From the study of the dynamic systems, this paper discussed the numerical method of the initial-boundary value problem of Rosenau-Burgers equation. It reveals the dissipation problems of nonlinear wave. By using the mesh method in time and space domain of the equation, a new implicit finite difference scheme of three levels was proposed. And the prior estimate of the solutions was obtained. It was proved that the finite difference scheme is convergent and stable. The numerical experiment indicates that the scheme is available and it is easy to implement, and computational time can be economized. The weighted difference scheme has universal significance and it is worth popularizing.

Key words: convergence, Rosenau-Burgers equation, finite difference scheme, stability

CLC Number:

O 241.82

SHAO Xin-Hui, XUE Guan-Yu, SHEN Hai-Long. A New Finite Difference Method for Rosenau-Burgers Equation[J]. Journal of Shanghai Jiaotong University, 2012, 46(10): 1693-1696.

References

［1］Rosenau P. A quasicontinuous description of a nonlinear transmission line［J］. Physica Scripta, 1986, 34(8): 827829.
［2］Rosenau P. Dynamics of dense discrete systems［J］. Progress of Theoretical Physics, 1988, 79(9): 10281042.
［3］Park M A. On the Rosenau equation［J］. Computation and Applied Mathematics, 1990, 9(2): 145152.
［4］肖龙飞, 杨建民, 于洋. 水波传播的无网格数值模拟［J］. 上海交通大学学报, 2007, 41(9): 15331537.
XIAO Longfei, YANG Jianmin, YU Yang. The numerical simulation on water wave propagation by meshless method［J］. Journal of Shanghai Jiaotong University, 200, 41(9): 15331537.
［5］LIU Liping, MEI Ming. A better asymptotic profile of RosenauBurgers equation［J］. Applied Mathematics and Computation, 2002, 131(1): 147170.
［6］LIU Liping, MEI Ming, WONG Yaushu. Asymptotic behavior of solutions to the RosenauBurgers equation with a periodic initial boundary［J］. Nonlinear Analysis, 2007, 67(8): 25272539.
［7］MA Weiyuan, YANG Aili, WANG Yang. A secondorder accurate linearized difference scheme for the RosenauBurgers equation［J］. Journal of Information & Computational Science, 2010, 7(8): 17931800.
［8］刘艳, 阿布都热西提·阿布都外力. RosenauBurgers方程一种新的数值解法［J］. 工程数学学报, 2011, 28(5): 665670.
LIU Yan, Abdurishit Abduwali. A new numerical method for RosenauBurgers equation［J］. Chinese Journal of Engineering Mathematics, 2011, 28(5): 665670.
［9］HU Bing, XU Youcai, HU Jinsong. CrankNicolson finite difference scheme for the RosenauBurgers equation［J］. Applied Mathematics and Computation, 2008, 204(1): 311316.
［10］Khaled O, Faycal A, Talha A, et al. A new conservative finite difference scheme for the Rosenau equation［J］. Applied Mathematics and Computation, 2008, 201(12): 3543.

[1]	Li Guolin, Chen Tong, He Shaoying, Yin Debin. Input Mapping-Based Model Predictive Control with Event-Triggered Adaptive Strategy for Rigid-Soft Hybrid Manipulator [J]. J Shanghai Jiaotong Univ Sci, 2026, 31(1): 36-47.
[2]	JIA Zhichao, LIU Mingyue, GUO Lin. Study on the Dynamic Response Characteristics of a Cantilever Intake Pipe under Internal Flow [J]. Ocean Engineering Equipment and Technology, 2026, 13(1): 46-57.
[3]	LI Xiang, CHEN Siyuan, ZHANG Jun, KE Deping, GAO Jiemai, YANG Huanhuan. Physics-Informed Fast Transient Stability Assessment of Non-Fixed Length in Power Systems [J]. Journal of Shanghai Jiao Tong University, 2025, 59(7): 962-970.
[4]	SU Cheng, ZHAO Xiangtang, YAN Zengzhen, ZHAO Zhigang, MENG Jiadong. Load Stability Analysis of a Floating Multi-Robot Coordinated Towing System [J]. J Shanghai Jiaotong Univ Sci, 2025, 30(6): 1162-1170.
[5]	GAO Lei, MA Junchao, LÜ Jing, LIU Jianing, WANG Chenxu, CAI Xu. Oscillatory Stability Assessment of Renewable Power Systems Based on Frequency-Domain Modal Analysis [J]. Journal of Shanghai Jiao Tong University, 2025, 59(6): 821-835.
[6]	WANG Yuyang, ZHANG Chen, ZHANG Yu, WANG Yiming, XU Po, CAI Xu. Reactive Power-Voltage Droop Gain Online Tuning Method of Photovoltaic Inverters for Improvement of Stable Output Power Capability in Weak Grids [J]. Journal of Shanghai Jiao Tong University, 2025, 59(6): 845-856.
[7]	SI Wenjia, CHEN Junru, ZHANG Chenglin, LIU Muyang. Influence of DC-Bus Voltage on Synchronization Stability of Grid-Following Converters [J]. Journal of Shanghai Jiao Tong University, 2025, 59(3): 313-322.
[8]	LI Hongxin, ZHONG Zuhao, LU Yi, WEN Yunfeng. Optimization of Frequency Control Parameters of Wind Farms Considering Inertia Security Requirement [J]. Journal of Shanghai Jiao Tong University, 2025, 59(3): 323-332.
[9]	ZONG Haoxiang, ZHANG Chen, BAO Yanhong, WU Feng, CAI Xu. Frequency-Domain Modeling and Synchronization Perspective Interaction Mechanism of GFL-GFM Converter System [J]. Journal of Shanghai Jiao Tong University, 2025, 59(2): 151-164.
[10]	HU Meng1, LIANG Zhenyuan2, XU Zhisheng3, CAO Yingju3. The Application of the Magnetizing Inrush Current Suppressor on the Nanhai Fenjin FPSO [J]. Ocean Engineering Equipment and Technology, 2025, 12(2): 48-51.
[11]	HAN Hualing, JIA Yichao, MA Zihan, DENG Jun, HUANG Meng. Stability Analysis of LCL Grid-Connected Inverter Based on Neural Network [J]. Journal of Shanghai Jiao Tong University, 2025, 59(12): 1805-1814.
[12]	FEI Renxiang, XU Hailiang, GE Pingjuan, CHEN Xiangyu. Assessment and Extension Method for Stable Operation Domain of DFIG in Asymmetric Weak Grid [J]. Journal of Shanghai Jiao Tong University, 2025, 59(10): 1510-1522.
[13]	DENG Xu^a (邓旭), FENG Zhengping^{a, b∗} (冯正平), HE Chenlu^a (何晨璐), CUI Zhenhua^a (崔振华). Attitude Stabilization of Unmanned Underwater Vehicle During Payloads Release [J]. J Shanghai Jiaotong Univ Sci, 2024, 29(5): 766-772.
[14]	ZHANG Xiaotian¹(张啸天), HE Defeng^1* (何德峰), LIAO Fei² (廖飞). Iterative Model Predictive Control for Automatic Carrier Landing of Carrier-Based Aircrafts Under Complex Surroundings and Constraints [J]. J Shanghai Jiaotong Univ Sci, 2024, 29(4): 712-724.
[15]	LI Boqun, SUN Zhifeng. A Coronavirus Herd Immunity Optimizer Based on Swarm Division [J]. Journal of Shanghai Jiao Tong University, 2024, 58(4): 555-564.