Symplectic Partitioned Runge–Kutta Methods for Constrained Hamiltonian Systems

Abstract

This article deals with the numerical treatment of Hamiltonian systems with holonomic constraints. A class of partitioned Runge–Kutta methods, consisting of the couples of s-stage Lobatto IIIA and Lobatto IIIB methods, has been discovered to solve these problems efficiently. These methods are symplectic, preserve all un-derlying constraints, and are superconvergent with order $2s - 2$. For separable Hamiltonians of the form $H(q,p) = \frac{1}{2}p^T M^{ - 1} p + U(q)$ the Rattle algorithm based on the Verlet method was up to now the only known symplectic method preserving the constraints. In fact this method turns out to be equivalent to the 2-stage Lobatto IIIA–IIIB method of order 2. Numerical examples have been performed which illustrate the theoretical results.