• 论文 •

### 关于辛算法稳定性的若干注记

1. 1 HLM, 中国科学院数学与系统科学研究院, 数学研究所, 北京 100190;
2 中国科学院大学数学科学学院, 北京 100049;
3 吉林大学 数学学院, 长春 130012
• 收稿日期:2020-08-11 出版日期:2020-11-15 发布日期:2020-11-15
• 基金资助:

国家自然科学基金~（11671392）资助.

Shang Zaijiu, Song Lina. SOME NOTES ON THE STABILITY OF SYMPLECTIC METHODS[J]. Mathematica Numerica Sinica, 2020, 42(4): 405-418.

### SOME NOTES ON THE STABILITY OF SYMPLECTIC METHODS

Shang Zaijiu1,2, Song Lina3

1. 1 HLM, Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
2 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China;
3 School of Mathematics, Jilin University, Changchun 130012, China
• Received:2020-08-11 Online:2020-11-15 Published:2020-11-15

In this paper we discuss the linear stability and nonlinear stability of symplectic methods. We illustrate the importance of studying these two types of stability in view of dynamics and its numerical computation and give a brief summary of some relevant results. We give a definition to the notion “analytic method” and show that an analytic symplectic method (e.g., Runge-Kutta symplectic methods) is absolutely linear stable if the stability function of the method is meromorphic on the complex plane. We notice that there are not only analytic methods (e.g., Runge-Kutta methods) but also non-analytic methods (e.g., various exponential integration methods based on constant variational formula) with absolutely linear stability. We review and discuss the main results, initiated by R. I. MacLachlan and S. K. Gray then further developed by S. Blanes, F. Casas and A. Murua, on the linear stability of splitting methods as well as on the construction of arbitrarily high order splitting symplectic methods and more efficient conjugate processed integrators with optimal linear stability by suitably choosing stability polynomial functions. Such optimized integrators show good numerical stability for linear dominated problems with weak nonlinear perturbations such as highly oscillatory systems and wave equations. We discuss the known results on nonlinear stability of symplectic methods by analyzing the stability of elliptic equilibrium, the exponentially slow diffusion of energy surface, and the preservation of the KAM invariant tori. At last we propose a new nonlinear stability notion by analyzing the homoclinic trajectories of the nonlinear oscillator of one degree of freedom on the basis of backward error analysis, to give a practically useful nonlinear stability criterion of symplectic methods.

MR(2010)主题分类:

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