### TWO NOVEL CLASSES OF ARBITRARY HIGH-ORDER STRUCTURE-PRESERVING ALGORITHMS FOR CANONICAL HAMILTONIAN SYSTEMS

Yonghui Bo1, Wenjun Cai2, Yushun Wang2

1. 1. Department of Mathematics, Anhui Normal University, Wuhu 241000, China;
2. Jiangsu Key Laboratory for NSLSCS, Jiangsu Collaborative Innovation Center of Biomedial Functional Materials, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
• Received:2021-03-15 Revised:2021-05-27 Published:2023-03-14
• Contact: Yushun Wang, Email:wangyushun@njnu.edu.cn
• Supported by:
This work is supported by the National Key Research and Development Project of China (Grant No. 2018YFC1504205), the National Natural Science Foundation of China (Grant No. 11771213, 11971242), the Major Projects of Natural Sciences of University in Jiangsu Province of China (Grant No. 18KJA110003) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.

Yonghui Bo, Wenjun Cai, Yushun Wang. TWO NOVEL CLASSES OF ARBITRARY HIGH-ORDER STRUCTURE-PRESERVING ALGORITHMS FOR CANONICAL HAMILTONIAN SYSTEMS[J]. Journal of Computational Mathematics, 2023, 41(3): 395-414.

In this paper, we systematically construct two classes of structure-preserving schemes with arbitrary order of accuracy for canonical Hamiltonian systems. The one class is the symplectic scheme, which contains two new families of parameterized symplectic schemes that are derived by basing on the generating function method and the symmetric composition method, respectively. Each member in these schemes is symplectic for any fixed parameter. A more general form of generating functions is introduced, which generalizes the three classical generating functions that are widely used to construct symplectic algorithms. The other class is a novel family of energy and quadratic invariants preserving schemes, which is devised by adjusting the parameter in parameterized symplectic schemes to guarantee energy conservation at each time step. The existence of the solutions of these schemes is verified. Numerical experiments demonstrate the theoretical analysis and conservation of the proposed schemes.

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