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Energy preserving methods on Riemannian manifolds

Celledoni, Elena; Eidnes, Sølve; Owren, Brynjulf; Ringholm, Torbjørn
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URI
http://hdl.handle.net/11250/2581726
Date
2018
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  • Institutt for matematiske fag [1390]
  • Publikasjoner fra CRIStin - NTNU [19694]
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.
Publisher
Cornell University
Journal
arXiv.org

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