Order Conditions for Canonical Runge–Kutta Schemes

Abstract

When numerically integrating Hamiltonian systems of differential equations, it is often advantageous to use canonical methods, i.e., methods that preserve the symplectic structure of the phase space, thus reproducing an important feature of the Hamiltonian flow. An s-stage Runge–Kutta (RK) method without redundant stages is canonical if and only if, with a standard notation, $b_i a_{ij} + b_j a_{ji} - b_i b_j = 0,1 \leqq i,j \leqq s$. It is shown that for canonical RK methods there are many redundancies in the standard order conditions. For a canonical method to have order p it is sufficient that the $b_i$’s, $a_{ij}$’s satisfy a system of algebraicequations that has, roughly speaking, an equation per nonrooted tree of order $\leqq p$. Furthermore, a new methodology is presented for the investigation of the order of canonical integration methods (not necessarily RK methods) when applied to Hamiltonian systems. In the new approach consistency is studied by Taylor expanding a suitable scalar function in ter...