Energy preserving methods on Riemannian manifolds

TL;DR

This paper extends energy-preserving discrete gradient methods to Riemannian manifolds, creating intrinsic schemes that maintain energy conservation without relying on coordinate choices, with applications demonstrated on spin systems.

Contribution

It introduces a general framework for energy-preserving methods on Riemannian manifolds, including higher-order schemes and error analysis, broadening the applicability of discrete gradient methods.

Findings

Methods are intrinsic and coordinate-independent.

Higher-order schemes can be constructed via collocation.

Numerical results demonstrate effectiveness on spin systems.

Abstract

The energy preserving discrete gradient methods are generalized to finite-dimensional Riemannian manifolds by definition of a discrete approximation to the Riemannian gradient, a retraction, and a coordinate center function. The resulting schemes are intrinsic and do not depend on a particular choice of coordinates, nor on embedding of the manifold in a Euclidean space. Generalizations of well-known discrete gradient methods, such as the average vector field method and the Itoh--Abe method are obtained. It is shown how methods of higher order can be constructed via a collocation-like approach. Local and global error bounds are derived in terms of the Riemannian distance function and the Levi-Civita connection. Some numerical results on spin system problems are presented.