Energy preserving methods on Riemannian manifolds
This work provides a theoretical framework for energy-preserving integration on manifolds, which is important for geometric numerical integration but is incremental as it extends existing methods.
The authors generalize energy-preserving discrete gradient methods to finite-dimensional Riemannian manifolds, obtaining intrinsic schemes like the average vector field and Itoh-Abe methods, with error bounds in terms of Riemannian distance. Numerical results on spin systems are presented.
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.