Energy and Quadratic Invariants Preserving Methods for Hamiltonian Systems with Holonomic Constraints

Abstract

We introduce a new class of parametrized structure-preserving partitioned Runge-Kutta (α-PRK) methods for Hamiltonian systems with holonomic constraints. The methods are symplectic for any fixed scalar parameter α, and are reduced to the usual symplectic PRK methods like Shake-Rattle method or PRK schemes based on Lobatto IIIA-IIIB pairs when α = 0. We provide a new variational formulation for symplectic PRK schemes and use it to prove that the α-PRK methods can preserve the quadratic invariants for Hamiltonian systems subject to holonomic constraints. Meanwhile, for any given consistent initial values (p0, q0) and small step size h > 0, it is proved that there exists α * = α(h, p0, q0) such that the Hamiltonian energy can also be exactly preserved at each step. Based on this, we propose some energy and quadratic invariants preserving α-PRK methods. These α-PRK methods are shown to have the same convergence rate as the usual PRK methods and perform very well in various numerical experiments.

Arbitrarily High-Order Energy-Conserving Methods for Hamiltonian Problems with Quadratic Holonomic Constraints

“…Among the proposed approaches, we mention: the popular Shake-Rattle method [1,37], later proved to be symplectic [30], higher order methods obtained via symplectic PRK methods [24], composition methods [35,36], symmetric LMFs [18], methods based on discrete derivatives [19], local parametrizations of the manifold containing the solution [3], projection techniques [38,41]. As additional references, we refer to [20,28,31,32,39] and to the monographs [4,21,23,29], and references therein. In particular, in [31] different methods, such as penalty methods, are analyzed and compared, whereas in [32], following a route akin to [13], an extra parameter is added to the methods to enforce the constraints.…”

“…As additional references, we refer to [20,28,31,32,39] and to the monographs [4,21,23,29], and references therein. In particular, in [31] different methods, such as penalty methods, are analyzed and compared, whereas in [32], following a route akin to [13], an extra parameter is added to the methods to enforce the constraints.…”