Order theory for discrete gradient methods
Abstract
The discrete gradient methods are integrators designed to preserve invariants of ordinary differential equations. From a formal series expansion of a subclass of these methods, we derive conditions for arbitrarily high order. We derive specific results for the average vector field discrete gradient, from which we get P-series methods in the general case, and B-series methods for canonical Hamiltonian systems. Higher order schemes are presented, and their applications are demonstrated on the H\'enon-Heiles system and a Lotka-Volterra system, and on both the training and integration of a pendulum system learned from data by a neural network.
Cite
@article{arxiv.2003.08267,
title = {Order theory for discrete gradient methods},
author = {Sølve Eidnes},
journal= {arXiv preprint arXiv:2003.08267},
year = {2022}
}
Comments
49 pages, 7 figures; v3: Corrected typos and minor mistakes, added application to Hamiltonian neural networks; v4: Minor corrections