Time Integration and Discrete Hamiltonian Systems

Abstract

Summary. This paper develops a formalism for the design of conserving time-integration schemes for Hamiltonian systems with symmetry. The main result is that, through the introduction of a discrete directional derivative, implicit second-order conserving schemes can be constructed for general systems which preserve the Hamiltonian along with a certain class of other first integrals arising from affine symmetries. Discrete Hamiltonian systems are introduced as formal abstractions of conserving schemes and are analyzed within the context of discrete dynamical systems; in particular, various symmetry and stability properties are investigated. Background and MotivationFirst integrals or conservation laws for Hamiltonian systems with symmetry are typically lost under numerical integration in time. In some cases, failure to maintain certain conservation laws can lead to physically impossible solutions [3], and in other cases to numerical instability [7], [21]- [24]. For Hamiltonian systems with symmetry it is thus generally desirable that numerical time-integration schemes preserve physically meaningful integrals from the underlying system. These types of integrators are usually referred to as conserving integrators and are the subject of this investigation. This paper develops a formalism for the design of conserving time-integration schemes for Hamiltonian systems with symmetry. The main result is that, through the introduction of a discrete directional derivative, implicit second-order conserving schemes can be constructed for general systems which preserve the Hamiltonian along with quadratic integrals arising from affine symmetries. Discrete Hamiltonian systems are introduced as formal abstractions of conserving schemes and are analyzed within the context of discrete dynamical systems; in particular, various symmetry and stability properties are 1 This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan C. Simo.

A mortar method for energy‐momentum conserving schemes in frictionless dynamic contact problems

“…For the two-body contact problem under consideration the configuration of the complete semidiscrete system is given by q(t) = q (1) (t) q (2) (t)…”

“…Following Puso and Laursen [7,Section 2], the potential function of the normal contact constraints pertaining to the mortar method can be derived from an integral form of the contact complementarity condition. Accordingly, (1) , t)·(u (1),h (X (1) , t)−u (2),h (X (2) , t)) d =:…”

“…In particular, we aim at conserving time-stepping schemes for contact-impact problems. To achieve this, we make use of the notion of a discrete gradient in the sense of Gonzalez [1,2] for the evaluation of the discrete contact forces. In Betsch and Hesch [3], this approach has been successfully applied to the node-to-segment (NTS) contact description.…”

Symplectic-energy-momentum preserving variational integrators

“…1, then one can have all three of these properties. Papers typified by Simo and Tarnow, 10 Simo, Tarnow, and Wong, 11 and Gonzalez 12 have focused on energy preserving algorithms, but they presumably fail ͑except, perhaps, in special cases, such as integrable systems͒ to be symplectic. Other approaches based on Hamilton's principle are those of Shibberu 13 and Lewis.…”

Rigid body dynamics with a scalable body, quaternions and perfect constraints

“…The smaller set of coordinates can be of advantage when no techniques can be applied to reduce the set of coordinates after discretization. For further details on energy consistent integrators, the reader is referred to [2,3,6,11].…”