Order Properties and Construction of Symplectic Runge-Kutta Methods (Journal of Computational Mathematics, 2021)
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 $1 \le p \le s, 0 \le l \le 2, s \le 2p+l \le 2s.$
Keywords: Numerical analysis; Symplectic Runge-Kutta methods; Simplifying conditions; Order results.