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 - <p style="text-align: justify;">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 $1 \le p \le s, 0 \le l \le 2, s \le 2p+l \le 2s.$&nbsp;&nbsp;</p>