Order Conditions for Canonical Runge–Kutta Schemes

Partitioned Runge-Kutta Methods for Separable Hamiltonian Problems

“…In later developments we shall need the generating function of a canonical PRK method [1,10,12]. By definition, this is a function S = S(p, q* ; A) such that the relations (2.4) p* = dS/dq*, q = dS/dp reproduce the equations (1.6) that define the time-step (p, q) >-> (p*, q*) of the method.…”

“…However, some bicolor trees are their own photographic negative, like ßx2y x or /?t4j3 . It is not difficult to see that the latter occurrence takes place if and only if ßx arises from coloring the vertices of what in [12] was called a superfluous (uncolored, unrooted) tree.…”

“…(Our treatment of the graph theory we need will be sketchy and based on pictorial representations. A rigorous treatment, like that given in [12], would result in too long a paper. )…”

“…The equivalence between (i) and (iii) is established next. We need the identity (3.5) If the (uncolored) tree x obtained from ßx by disregarding the coloring of the vertices is superfluous, then (3.5) has been proved in [12,Theorem 4.5]. If x is superfluous, then (3.5) is trivial.…”

“…The well-known RK schemes based on Gauss-Legendre quadrature are canonical [9]. Diagonally implicit methods satisfying (1.3) are derived in [12] and tested in [4]. On the other hand, it is easy to see that (1.3) implies that (1.2) cannot be explicit.…”