Energy preserving methods on Riemannian manifolds

2018/05/19 by Celledoni, Elena, Eidnes, Sølve, Owren, Brynjulf +1 · 2 citations
#37K05 #53B99 #65L05 #82-08 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1805.07578

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.

Cited by

• Shortest Geodesic Loops, Sectional Curvature, and Injectivity Radius of the Stiefel Manifold
• A Unified Discrete Gradient-SAV Framework for Structure-Preserving Integration

Related

• Dissipative numerical schemes on Riemannian manifolds with applications to gradient flows
• Riemannian Newton methods for energy minimization problems of Kohn-Sham type
• Energy-adaptive Riemannian optimization on the Stiefel manifold
• Higher order discrete gradient method by the discontinuous Galerkin time-stepping method
• From Nesterov's Estimate Sequence to Riemannian Acceleration