辛方法的校正公式

1996年Wisdom等提出了对辛方法进行校正的概念和实践，现在继续对辛校正进行详尽讨论和数值比较，尤其对哈密顿函数可分解为一个主要部分和多个次要部分的一般情形，用Lie级数推导任意阶的各种辛算法的一次和二次辛校正公式并对一些算法给出具体的辛校正公式。又以日、木、土三体问题为模型进行数值实验，结果表明一次辛校正能提高精度，改善数值稳定性。计算效率也比较高，因而值得推荐使用，辛方法通常用大步长数值积分，这时二次辛校正并没有显著提高结果的精度，却大大增加了计算时间，不应予以推荐。

On Correctors of Symplectic Integrators

In 1996, Wisdom et al proposed the concept of corrector for a symplectic integrator and put it into practice. This is an intensive discussion with numerical comparison on symplectic correctors. Then it gives the method to derive the first and second correctors of any symplectic integrators by Lie series in a general case, in which the Hamiltonian can be separated into a main integrable part and several smaller integrable parts. Numerical experiments have been taken in a Sun-Jupiter-Saturn 3-body problem. It has shown that the first order corrector can raise precision, improve numerical stability, and has a good computer efficiency. Therefore, it deserves to be recommended. A large stepsize is usually adopted in the application of symplectic integrators. In this case the second order corrector has no notable advantages in raising precision. Furthermore, they would take much more computational time and should not be recommended.