On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems

Abstract

In this paper we generalize the class of explicit partitioned Runge-Kutta (PRK) methods for separable Hamiltonian systems to systems with holonomic constraints. For a convenient analysis of such schemes, we first generalize the backward error analysis for systems in ${\Bbb R}^m$ to systems on manifolds embedded in ${\Bbb R}^m$ . By applying this analysis to constrained PRK methods, we prove that such methods will, in general, suffer from order reduction as well-known for higher-index differential-algebraic equations. However, this order reduction can be avoided by a proper modification of the standard PRK methods. This modification increases the number of projection steps onto the constraint manifold but leaves the number of force evaluations constant. We also give a numerical comparison of several second, fourth, and sixth order methods.