几类辛方法的数值稳定性研究

On the Numerical Stability for Some Symplectic Integrators

主要对一阶隐式Euler辛方法M1、二阶隐式Euler中点辛方法M2、一阶显辛Euler方法M3和二阶leapfrog显辛积分器M4共4种辛方法及一些组合算法进行了通常意义下的线性稳定性分析．针对线性哈密顿系统,理论上找到每个数值方法的稳定区,然后用数值方法检验其正确性．对于哈密顿函数为实对称二次型的情况,为了理论推导便利,特推荐采用相似变换将二次型的矩阵对角化来研究辛方法的线性稳定性．当哈密顿分解为一个主要部分和一个小摄动次要部分且二者皆可积时,无论是线性系统还是非线性系统,这种主次分解与哈密顿具有动势能分解相比,明显扩大了辛方法的稳定步长范围．

This research deals mainly with an analysis of the linear stability of several symplectic integrators for a linear Hamiltonian system, which involve the first-order implicit symplectic Euler scheme, the second-order implicit centered Euler difference scheme, the first-order explicit symplectic Euler scheme and the second-order explicit leapfrog symplectic integrator. Meantime, a stable region for each integrator is found. The fact is also checked by numerical tests. Especially for a system with a real symmetric quadratic form, a simpler way to study the numerical stability is to use diagonalizing transformations. As an emphasis, a rather larger stable time step of each algorithm is admissible for either a linear or nonlinear system with integrable separations of one main piece and another petty piece rather than a kinetic energy and a potential energy.