Properties of Hamiltonian Variational Integrators

TL;DR

This paper analyzes the properties of Hamiltonian variational integrators, establishing their accuracy relations, introducing adjoint concepts, and examining resonance behaviors through numerical experiments.

Contribution

It provides a variational error analysis linking integrator accuracy to generating function approximation and introduces the notion of an adjoint discrete Hamiltonian.

Findings

Accuracy depends on how well generating functions approximate exact discrete Hamiltonians

The same one-step method and quadrature can produce different integrators when constructing discrete Lagrangians and Hamiltonians

Resonance behavior varies with the type of generating functions used and relates to boundary-value problem ill-posedness

Abstract

Discrete Hamiltonian variational integrators are derived from Type II and Type III generating functions for symplectic maps, and in this paper we establish a variational error analysis result that relates the order of accuracy of the associated numerical methods with the extent to which these generating functions approximate the exact discrete Hamiltonians. We also introduce the notion of an adjoint discrete Hamiltonian, and relate it to the adjoint of the associated symplectic integrator. We show that when constructing discrete Lagrangians and discrete Hamiltonians using the Taylor variational integrator approach, the same underlying one-step method and quadrature rule does not necessarily lead to the same symplectic integrator, and the same observation holds when developing variational integrators based on averaging techniques. Numerical experiments also indicate that the resonance behavior of variational integrators also depend on the type of generating functions used, and we relate this resonance behavior to the ill-posedness of the boundary-value problems used to define the exact discrete Lagrangian and exact discrete Hamiltonian.