Energy-preserving methods on Riemannian manifolds

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.