Superconvergence of Galerkin variational integrators

TL;DR

This paper proves that Galerkin variational integrators achieve a convergence order of twice the polynomial degree used, provided the quadrature rule is sufficiently accurate, advancing the understanding of their numerical performance.

Contribution

It establishes a theoretical convergence order of 2s for Galerkin variational integrators with sufficiently accurate quadrature rules.

Findings

Order of convergence is 2s for Galerkin variational integrators.

Quadrature rule accuracy is crucial for achieving superconvergence.

Provides theoretical foundation for the effectiveness of these integrators.

Abstract

We study the order of convergence of Galerkin variational integrators for ordinary differential equations. Galerkin variational integrators approximate a variational (Lagrangian) problem by restricting the space of curves to the set of polynomials of degree at most $s$ and approximating the action integral using a quadrature rule. We show that, if the quadrature rule is sufficiently accurate, the order of the integrators thus obtained is $2s$.