Prolongation-Collocation Variational Integrators

TL;DR

This paper presents a new method for creating higher-order variational integrators for Hamiltonian systems, utilizing Hermite interpolation and Euler-Maclaurin quadrature to achieve smooth solutions with improved accuracy.

Contribution

It introduces a novel construction of variational integrators using Hermite polynomials and collocation, enhancing the order and smoothness of solutions for Hamiltonian ODEs.

Findings

The proposed integrators achieve higher order accuracy.

Performance comparison shows improved approximation properties.

The method effectively generates globally smooth solutions.

Abstract

We introduce a novel technique for constructing higher-order variational integrators for Hamiltonian systems of ODEs. In particular, we are concerned with generating globally smooth approximations to solutions of a Hamiltonian system. Our construction of the discrete Lagrangian adopts Hermite interpolation polynomials and the Euler-Maclaurin quadrature formula, and involves applying collocation to the Euler-Lagrange equation and its prolongation. Considerable attention is devoted to the order analysis of the resulting variational integrators in terms of approximation properties of the Hermite polynomials and quadrature errors. A performance comparison is presented on a selection of these integrators.