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arXiv · 1805.07578

Energy preserving methods on Riemannian manifolds

Elena Celledoni·Sølve Eidnes·Brynjulf Owren·Torbjørn Ringholm
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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.

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Elena Celledoni, Sølve Eidnes, Brynjulf Owren, Torbjørn Ringholm. 2018-05-19. Energy preserving methods on Riemannian manifolds. https://arxiv.org/abs/1805.07578

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