Energy and momentum conserving methods of arbitrary order for the numerical integration of equations of motion

Abstract

Summary. In Part I of this work, numerical methods were derived for the solution of the equations of motion of a single particle subject to a central force which conserved exactly the energy and momenta. In the present work, the methodology of Part I is extended, in part, to motion of a system of particles, in that the energy and linear momentum are conserved exactly. In addition, the angular momentum will be conserved to one more order of accuracy than in conventional methods. Exact conservation of the total angular momentum results only for the lowest order numerical approximation, which is equivalent to the "discrete mechanics" presented elsewhere.

Exactly conservative integrators

“…This indicates that our restriction of the general constraint (13) to the condition (14) merely ensures that the modal energies evolve in a manner consistent with the Euler discretization of the energy equations. Below we will give derivations of integrators for other systems based on this idea.…”

Conservative fourth-order time integration of non-linear dynamic systems

“…In the last decades there has been a considerable interest in the development of time integration methods that conserve invariants of the problem like energy and momentum. An early contribution is the series of papers by LaBudde and Greenspan [1,2], who developed discrete time integration methods for particle dynamics based on energy and momentum conservation. This type of approach to the numerical calculation of nonlinear dynamic response was further developed by Simo et al [3,4,5] covering rigid-body motion, elastodynamics and general Hamiltonian systems.…”

Energy Conservation for the Simulation of Deformable Bodies

“…Given the importance of energy conservation, we begin our exposition in Section 2 with a bottom-up approach to energy conservation for non-linear elastic systems similar to some of the earliest papers on the topic, such as [15] which considered the motion of a single particle and the subsequent paper [16] that considered systems of particles. This leads us one step at a time towards a conclusion similar to theirs, that an iterative approach is required (one of their methods requires iteration and the other requires dual iteration).…”