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Volume 13, Issue 1
On Stability of Symplectic Algorithms

Wang-Yao Li

J. Comp. Math., 13 (1995), pp. 64-69.

Published online: 1995-02

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  • Abstract

The stability of symplectic algorithms is discussed in this paper. There are following conclusions.
1. Symplectic Runge-Kutta methods and symplectic one-step methods with high order derivative are unconditionally critically stable for Hamiltonian systems. Only some of them are A-stable for non-Hamiltonian systems. The criterion of judging A-stability is given.
2. The hopscotch schemes are conditionally critically stable for Hamiltonian systems. Their stability regions are only a segment on the imaginary axis for non-Hamiltonian systems.
3. All linear symplectic multistep methods are conditionally critically stable except the trapezoidal formula which is unconditionally critically stable for Hamiltonian systems. Only the trapezoidal formula is A-stable, and others only have segments on the imaginary axis as their stability regions for non-Hamiltonian systems.  

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@Article{JCM-13-64, author = {Li , Wang-Yao}, title = {On Stability of Symplectic Algorithms}, journal = {Journal of Computational Mathematics}, year = {1995}, volume = {13}, number = {1}, pages = {64--69}, abstract = {

The stability of symplectic algorithms is discussed in this paper. There are following conclusions.
1. Symplectic Runge-Kutta methods and symplectic one-step methods with high order derivative are unconditionally critically stable for Hamiltonian systems. Only some of them are A-stable for non-Hamiltonian systems. The criterion of judging A-stability is given.
2. The hopscotch schemes are conditionally critically stable for Hamiltonian systems. Their stability regions are only a segment on the imaginary axis for non-Hamiltonian systems.
3. All linear symplectic multistep methods are conditionally critically stable except the trapezoidal formula which is unconditionally critically stable for Hamiltonian systems. Only the trapezoidal formula is A-stable, and others only have segments on the imaginary axis as their stability regions for non-Hamiltonian systems.  

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9251.html} }
TY - JOUR T1 - On Stability of Symplectic Algorithms AU - Li , Wang-Yao JO - Journal of Computational Mathematics VL - 1 SP - 64 EP - 69 PY - 1995 DA - 1995/02 SN - 13 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9251.html KW - AB -

The stability of symplectic algorithms is discussed in this paper. There are following conclusions.
1. Symplectic Runge-Kutta methods and symplectic one-step methods with high order derivative are unconditionally critically stable for Hamiltonian systems. Only some of them are A-stable for non-Hamiltonian systems. The criterion of judging A-stability is given.
2. The hopscotch schemes are conditionally critically stable for Hamiltonian systems. Their stability regions are only a segment on the imaginary axis for non-Hamiltonian systems.
3. All linear symplectic multistep methods are conditionally critically stable except the trapezoidal formula which is unconditionally critically stable for Hamiltonian systems. Only the trapezoidal formula is A-stable, and others only have segments on the imaginary axis as their stability regions for non-Hamiltonian systems.  

Li , Wang-Yao. (1995). On Stability of Symplectic Algorithms. Journal of Computational Mathematics. 13 (1). 64-69. doi:
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