On symplectic and symmetric ARKN methods
- 作者：Wei Shi shuier628@163.com ; Xinyuan Wu ; xywu@nju.edu.cn
- 关键词：Oscillatory second-order initial value problems ; ARKN integrators ; Symplecticness conditions ; Oscillatory system ; Hamiltonian system ; Symmetry
- 刊名：Computer Physics Communications
- 出版年：2012
- 期刊代码：30_00104655
- 类别：cp
- 出版时间：June, 2012
- 卷：183
- 期：6
- 页码：1250-1258
- 文件大小：396 K
摘要
Symplecticness and symmetry are favorable properties for solving Hamiltonian systems. For the oscillatory second-order initial value problems of the form , adapted Runge-Kutta-Nystr枚m methods (ARKN methods, in short notation) were investigated by several authors. In a wide range of physical applications from molecular dynamics to nonlinear wave propagation, an important class of the problems is Hamiltonian systems for which symplectic methods should be preferred. Hence it is quite natural to raise a question of the symplecticness for ARKN methods. In this paper we investigate the symplecticness conditions of ARKN methods for separable Hamiltonian systems. We conclude that there exist only one-stage explicit symplectic ARKN (SARKN, in short notation) methods under the symplecticness conditions of ARKN methods. The SARKN methods have a special form and the algebraic order cannot exceed 2. We also point out that no ARKN method can be symmetric. An explicit SARKN method of order two is proposed with the analysis of phase and stability properties. The numerical results accompanied show good performance for the new explicit symplectic algorithm in comparison with the popular symplectic methods in the scientific literature.