李级数算法和显式辛算法的相位分析

Phase analysis of Lie series algorithm and explicit symplectic algorithm

以线性可分Hamilton动力学系统为例,研究了李级数算法和显式辛算法的相位精度,研究了李级数算法的保辛精度及其保辛精度的提高方法;指出了显式辛算法相位精度与算法阶次的不协调性,即辛算法的阶次高并不意味着其相位精度也高,李级数算法不存在这种问题,指出了一个算法的相位可能超前也可能滞后。分析结果表明三阶显式辛算法具有比较高的相位精度。

Taking a linear separable Hamiltonian system as an example, the phase errors of Lie series algorithms and explicit symplectic geometric algorithms were analyzed in details, the accuracy order for amplitude preserving symplectic of Lie series algorithms and its improving method were investigated, by which the amplitude accuracy is increased and but phase accuracy is effected less. There is one conclusion that the order of explicit symplectic integration algorithm is about the amplitude not about the phase, but the order of Lie series algorithms is about the amplitude and the phase simultaneously. The phase accuracy of the third order explicit symplectic method is higher than that of the fourth order ex- plicit and implicit symplectic integration algorithms, and that of the fourth order Lie series algorithm and the fourth order modified Lie series algorithm. Moreover, the phase of an algorithm can either backward or forward comparing with the exact solution. Finally, it is concluded for Lie series algorithm that the accuracy of amplitude and phase can be improved with the increase of its order, but the conclusion is not valid for the explicit symplectic integration methods. Taking the efficiency and the accuracy into account, the three order explicit symplectic method is superior to the others for linear and nonlinear separable Hamiltonian systems.