Family of symplectic implicit Runge-Kutta formulae
This paper is concerned with the construction of canonical Runge-Kutta (RK) methods, i.e. implicit RK methods which integrate Hamiltonian systems of ordinary differential equations preserving the symplectic structure of the phase space. Firstly, the authors show that the canonicity conditions [see e.g. \textit{J. M. Sanz-Serna}, BIT 28, No. 4, 877-883 (1988; Zbl 0655.70013)] of an \(s\)-stages method defined by its Butcher tableau \(A=(a_{ij})\), \(b=(b_ i)\), \((1\leq i,j\leq s)\) can be written as a linear system in the \(s^ 2\) coefficients \(a_{ij}\), proving that the dimension of its kernel is \((s- 1)(s-2)/2\). This fact implies that the set of \(s\)-stages canonical RK methods depends on \((s-1)(s-2)/2\) parameters. Finally, taking into account the usual order conditions the construction of canonical methods with \(s\leq 3\) stages is studied.
scientific article; zbMATH DE number 93335