Spectral Variational Integrators

TL;DR

This paper introduces a new spectral variational integrator for Lagrangian mechanics that is symplectic, momentum-preserving, and converges optimally, with demonstrated numerical convergence rates.

Contribution

It develops a spectral variational integrator using Galerkin techniques that achieves geometric convergence and preserves geometric invariants.

Findings

Converges geometrically with optimal rate.

Preserves symplectic structure and momentum.

Numerical examples confirm theoretical convergence rates.

Abstract

In this paper, we present a new variational integrator for problems in Lagrangian mechanics. Using techniques from Galerkin variational integrators, we construct a scheme for numerical integration that converges geometrically, and is symplectic and momentum preserving. Furthermore, we prove that under appropriate assumptions, variational integrators constructed using Galerkin techniques will yield numerical methods that are in a certain sense optimal, converging at the same rate as the best possible approximation in a certain function space. We further prove that certain geometric invariants also converge at an optimal rate, and that the error associated with these geometric invariants is independent of the number of steps taken. We close with several numerical examples that demonstrate the predicted rates of convergence.