Invariant Variational Schemes for Ordinary Differential Equations

TL;DR

This paper introduces a new method for creating invariant variational schemes for ODEs using equivariant moving frames, leading to schemes that preserve symmetries and conservation laws, and outperform standard methods.

Contribution

The paper presents a novel approach to construct invariant variational schemes for ODEs via invariantization of discrete Lagrangians, ensuring symmetry preservation and improved numerical performance.

Findings

Invariant schemes preserve variational and divergence symmetries.

Schemes are exactly conservative due to Noether's theorem.

Numerical simulations show superior performance over standard discretizations.

Abstract

We propose a novel algorithmic method for constructing invariant variational schemes of systems of ordinary differential equations that are the Euler-Lagrange equations of a variational principle. The method is based on the invariantization of standard, non-invariant discrete Lagrangian functionals using equivariant moving frames. The invariant variational schemes are given by the Euler-Lagrange equations of the corresponding invariantized discrete Lagrangian functionals. We showcase this general method by constructing invariant variational schemes of ordinary differential equations that preserve variational and divergence symmetries of the associated continuous Lagrangians. Noether's theorem automatically implies that the resulting schemes are exactly conservative. Numerical simulations are carried out and show that these invariant variational schemes outperform standard numerical discretizations.