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Volume 18, Issue 6
Order Properties and Construction of Symplectic Runge-Kutta Methods

Shou-Fu Li

J. Comp. Math., 18 (2000), pp. 645-656.

Published online: 2000-12

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  • Abstract

The main results of this paper are as follows: (1) Suppose an s stage Runge-Kutta method is consistent, irreducible, non-confluent and symplectic. Then this method is of order at least $2p+l(1 \le p \le s-1)$ provided that the simplifying conditions $C(p)$ (or $D(p)$ with non-zero weights) and $B(2p+l)$ hold, where $l=0,1,2.$ (2) Suppose an s stage Runge-Kutta method is consistent, irreducible and non-confluent, and satisfies the simplifying conditions $C(p)$ and $D(p)$ with $0 < p \le s.$ Then this method is symplectic if and only if either $p = s$ or the nonlinear stablility matrix $M$ of the method has an $(s-p)×(s-p)$ chief submatrix $\hat{M}=0.$ (3) Using the results (1) and (2) as bases, we present a general approach for the construction of symplectic Runge-Kutta methods, and a software has benn designed, by means of which, the coefficients of s stage symplectic Runge-Kutta methods satisfying $C(p), D(p)$ and $B(2p+l)$ can be easily computed, where $1 \le p \le s, 0 \le l \le 2, s \le 2p+l \le 2s.$  

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@Article{JCM-18-645, author = {Li , Shou-Fu}, title = {Order Properties and Construction of Symplectic Runge-Kutta Methods}, journal = {Journal of Computational Mathematics}, year = {2000}, volume = {18}, number = {6}, pages = {645--656}, abstract = {

The main results of this paper are as follows: (1) Suppose an s stage Runge-Kutta method is consistent, irreducible, non-confluent and symplectic. Then this method is of order at least $2p+l(1 \le p \le s-1)$ provided that the simplifying conditions $C(p)$ (or $D(p)$ with non-zero weights) and $B(2p+l)$ hold, where $l=0,1,2.$ (2) Suppose an s stage Runge-Kutta method is consistent, irreducible and non-confluent, and satisfies the simplifying conditions $C(p)$ and $D(p)$ with $0 < p \le s.$ Then this method is symplectic if and only if either $p = s$ or the nonlinear stablility matrix $M$ of the method has an $(s-p)×(s-p)$ chief submatrix $\hat{M}=0.$ (3) Using the results (1) and (2) as bases, we present a general approach for the construction of symplectic Runge-Kutta methods, and a software has benn designed, by means of which, the coefficients of s stage symplectic Runge-Kutta methods satisfying $C(p), D(p)$ and $B(2p+l)$ can be easily computed, where $1 \le p \le s, 0 \le l \le 2, s \le 2p+l \le 2s.$  

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9075.html} }
TY - JOUR T1 - Order Properties and Construction of Symplectic Runge-Kutta Methods AU - Li , Shou-Fu JO - Journal of Computational Mathematics VL - 6 SP - 645 EP - 656 PY - 2000 DA - 2000/12 SN - 18 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9075.html KW - Numerical analysis, Symplectic Runge-Kutta methods, Simplifying conditions, Order results. AB -

The main results of this paper are as follows: (1) Suppose an s stage Runge-Kutta method is consistent, irreducible, non-confluent and symplectic. Then this method is of order at least $2p+l(1 \le p \le s-1)$ provided that the simplifying conditions $C(p)$ (or $D(p)$ with non-zero weights) and $B(2p+l)$ hold, where $l=0,1,2.$ (2) Suppose an s stage Runge-Kutta method is consistent, irreducible and non-confluent, and satisfies the simplifying conditions $C(p)$ and $D(p)$ with $0 < p \le s.$ Then this method is symplectic if and only if either $p = s$ or the nonlinear stablility matrix $M$ of the method has an $(s-p)×(s-p)$ chief submatrix $\hat{M}=0.$ (3) Using the results (1) and (2) as bases, we present a general approach for the construction of symplectic Runge-Kutta methods, and a software has benn designed, by means of which, the coefficients of s stage symplectic Runge-Kutta methods satisfying $C(p), D(p)$ and $B(2p+l)$ can be easily computed, where $1 \le p \le s, 0 \le l \le 2, s \le 2p+l \le 2s.$  

Li , Shou-Fu. (2000). Order Properties and Construction of Symplectic Runge-Kutta Methods. Journal of Computational Mathematics. 18 (6). 645-656. doi:
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