Invariant Variational Schemes for Ordinary Differential Equations

by   Alex Bihlo, et al.

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.


page 1

page 2

page 3

page 4


On the development of symmetry-preserving finite element schemes for ordinary differential equations

In this paper we introduce a procedure, based on the method of equivaria...

Discretization by euler's method for regular lagrangian flow

This paper is concerned with the numerical analysis of the explicit Eule...

Superconvergence of Galerkin variational integrators

We study the order of convergence of Galerkin variational integrators fo...

Learning Nonparametric Ordinary differential Equations: Application to Sparse and Noisy Data

Learning nonparametric systems of Ordinary Differential Equations (ODEs)...

Variational integrators for non-autonomous systems with applications to stabilization of multi-agent formations

Numerical methods that preserve geometric invariants of the system, such...

Performance of Borel-Laplace integrator for the resolution of stiff and non-stiff problems

A stability analysis of the Borel-Laplace series summation technique, us...