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Energy-preserving methods on Riemannian manifolds


Authors: Elena Celledoni, Sølve Eidnes, Brynjulf Owren and Torbjørn Ringholm
Journal: Math. Comp. 89 (2020), 699-716
MSC (2010): Primary 37K05; Secondary 53B99, 65L05
DOI: https://doi.org/10.1090/mcom/3470
Published electronically: September 6, 2019
MathSciNet review: 4044447
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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 formulated only in terms of these three objects and do not otherwise depend on a particular choice of coordinates or 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. Numerical results are presented, for problems on the two-sphere, the paraboloid, and the Stiefel manifold.


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Additional Information

Elena Celledoni
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, N–7491 Trondheim, Norway
Email: elena.celledoni@ntnu.no

Sølve Eidnes
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, N–7491 Trondheim, Norway
Email: solve.eidnes@ntnu.no

Brynjulf Owren
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, N–7491 Trondheim, Norway
Email: brynjulf.owren@ntnu.no

Torbjørn Ringholm
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, N–7491 Trondheim, Norway
Email: ringholm@gmail.com

DOI: https://doi.org/10.1090/mcom/3470
Keywords: Geometric integration, discrete gradients, Riemannian manifolds, numerical analysis
Received by editor(s): May 24, 2018
Received by editor(s) in revised form: January 17, 2019, and May 31, 2019
Published electronically: September 6, 2019
Additional Notes: This work was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 691070.
Article copyright: © Copyright 2019 American Mathematical Society