Construction and analysis of higher order Galerkin variational integrators

TL;DR

This paper develops higher order variational integrators using polynomial spaces and quadrature rules, improving accuracy and efficiency in structure-preserving mechanical system simulations.

Contribution

It introduces a new class of higher order Galerkin variational integrators with enhanced convergence and stability properties, extending previous methods.

Findings

Higher order schemes increase accuracy and reduce computational cost.

Numerical investigations confirm optimal convergence rates.

Constructed integrators preserve geometric structure and stability.

Abstract

In this work we derive and analyze variational integrators of higher order for the structure-preserving simulation of mechanical systems. The construction is based on a space of polynomials together with Gauss and Lobatto quadrature rules to approximate the relevant integrals in the variational principle. The use of higher order schemes increases the accuracy of the discrete solution and thereby decrease the computational cost while the preservation properties of the scheme are still guaranteed. The order of convergence of the resulting variational integrators are investigated numerically and it is discussed which combination of space of polynomials and quadrature rules provide optimal convergence rates. For particular integrators the order can be increased compared to the Galerkin variational integrators previously introduced in Marsden & West 2001. Furthermore, linear stability properties, time reversibility, structure-preserving properties as well as efficiency for the constructed variational integrators are investigated and demonstrated by numerical examples.