On Higher-order Semi-explicit Symplectic Partitioned Runge-kutta Methods for Constrained Hamiltonian Systems

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 rst generalize the backward error analysis for systems in I R m to systems on manifolds embedded in I R m. By applying this analysis to constrained PRK methods, we prove that such methods will, in general, suuer from order reduction as well-known for higher-index diierential-algebraic equations. However, this order reduction can be avoided by a proper modiication of the standard PRK methods. This modiication 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.