General Techniques for Constructing Variational Integrators

TL;DR

This paper reviews systematic techniques for constructing variational integrators, analyzing how choices in quadrature, function spaces, and methods affect their accuracy and conservation properties, supported by numerical examples.

Contribution

It provides a unified framework linking construction methods of variational integrators with their accuracy and conservation features, including new theoretical insights.

Findings

Quadrature choice influences integrator accuracy

Finite-dimensional spaces affect momentum conservation

Numerical examples validate theoretical results

Abstract

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi's solution of the Hamilton-Jacobi equation. These two characterizations lead to the Galerkin and shooting-based constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimensional function space or a one-step method. We prove that the properties of the quadrature formula, finite-dimensional function space, and underlying one-step method determine the order of accuracy and momentum-conservation properties of the associated variational integrators. We also illustrate these systematic methods for constructing variational integrators with numerical examples.