Volume 22, Issue 5
Symplectic RK Methods and Symplectic PRK Methods with Real Eigenvalues

Hong-yu Liu & Geng Sun

DOI:

J. Comp. Math., 22 (2004), pp. 769-776

Published online: 2004-10

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  • Abstract

Properties of symplectic Runge-Kutta (RK) methods and symplectic partitioned Runge- Kutta (PRK) methods with real eigenvalues are discussed in this paper. It is shown that an s stage such method can't reach order more than s + 1. Particularly, we prove that no symplectic RK method with real eigenvalues exists in stage s of order s + I when s is even. But an example constructed by using the W-transformation shows that PRK method of this type does not necessarily meet this order barrier. Another useful way other than W-transformation to construct symplectic PRK method with real eigenvalues is then presented. Finally, a class of efficient symplectic methods is recommended.

  • Keywords

Runge-Kutta method Partitioned Runge-Kutta method Symplectic Real eigenvalues

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@Article{JCM-22-769, author = {}, title = {Symplectic RK Methods and Symplectic PRK Methods with Real Eigenvalues}, journal = {Journal of Computational Mathematics}, year = {2004}, volume = {22}, number = {5}, pages = {769--776}, abstract = { Properties of symplectic Runge-Kutta (RK) methods and symplectic partitioned Runge- Kutta (PRK) methods with real eigenvalues are discussed in this paper. It is shown that an s stage such method can't reach order more than s + 1. Particularly, we prove that no symplectic RK method with real eigenvalues exists in stage s of order s + I when s is even. But an example constructed by using the W-transformation shows that PRK method of this type does not necessarily meet this order barrier. Another useful way other than W-transformation to construct symplectic PRK method with real eigenvalues is then presented. Finally, a class of efficient symplectic methods is recommended. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/10302.html} }
TY - JOUR T1 - Symplectic RK Methods and Symplectic PRK Methods with Real Eigenvalues JO - Journal of Computational Mathematics VL - 5 SP - 769 EP - 776 PY - 2004 DA - 2004/10 SN - 22 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/10302.html KW - Runge-Kutta method KW - Partitioned Runge-Kutta method KW - Symplectic KW - Real eigenvalues AB - Properties of symplectic Runge-Kutta (RK) methods and symplectic partitioned Runge- Kutta (PRK) methods with real eigenvalues are discussed in this paper. It is shown that an s stage such method can't reach order more than s + 1. Particularly, we prove that no symplectic RK method with real eigenvalues exists in stage s of order s + I when s is even. But an example constructed by using the W-transformation shows that PRK method of this type does not necessarily meet this order barrier. Another useful way other than W-transformation to construct symplectic PRK method with real eigenvalues is then presented. Finally, a class of efficient symplectic methods is recommended.
Hong-yu Liu & Geng Sun. (1970). Symplectic RK Methods and Symplectic PRK Methods with Real Eigenvalues. Journal of Computational Mathematics. 22 (5). 769-776. doi:
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