关于辛算法稳定性的若干注记

SOME NOTES ON THE STABILITY OF SYMPLECTIC METHODS

我们讨论辛算法的线性稳定性和非线性稳定性,从动力系统和计算的角度论述了研究辛算法的这两类稳定性问题的重要性,分析总结了相关重要结果.我们给出了解析方法的明确定义,证明了稳定函数是亚纯函数的解析辛方法是绝对线性稳定的.绝对线性稳定的辛方法既有解析方法(如Runge-Kutta辛方法),也有非解析方法(如基于常数变易公式对线性部分进行指数积分而对非线性部分使用其它数值积分的方法).我们特别回顾并讨论了R.I.McLachlan,S.K.Gray和S.Blanes,F.Casas,A.Murua等关于分裂算法的线性稳定性结果,如通过选取适当的稳定多项式函数构造具有最优线性稳定性的任意高阶分裂辛算法和高效共轭校正辛算法,这类经优化后的方法应用于诸如高振荡系统和波动方程等线性方程或者线性主导的弱非线性方程具有良好的数值稳定性.我们通过分析辛算法在保持椭圆平衡点的稳定性,能量面的指数长时间慢扩散和KAM不变环面的保持等三个方面阐述了辛算法的非线性稳定性,总结了相关已有结果.最后在向后误差分析基础上,基于一个自由度的非线性振子和同宿轨分析法讨论了辛算法的非线性稳定性,提出了一个新的非线性稳定性概念,目的是为辛算法提供一个实际可用的非线性稳定性判别法.

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.