B-stability of numerical integrators on Riemannian manifolds
Abstract
We propose a generalization of nonlinear stability of numerical one-step integrators to Riemannian manifolds in the spirit of Butcher's notion of B-stability. Taking inspiration from Simpson-Porco and Bullo, we introduce non-expansive systems on such manifolds and define B-stability of integrators. In this first exposition, we provide concrete results for a geodesic version of the Implicit Euler (GIE) scheme. We prove that the GIE method is B-stable on Riemannian manifolds with non-positive sectional curvature. We show through numerical examples that the GIE method is expansive when applied to a certain non-expansive vector field on the 2-sphere, and that the GIE method does not necessarily possess a unique solution for large enough step sizes. Finally, we derive a new improved global error estimate for general Lie group integrators.
Keywords
Cite
@article{arxiv.2308.08261,
title = {B-stability of numerical integrators on Riemannian manifolds},
author = {Martin Arnold and Elena Celledoni and Ergys Çokaj and Brynjulf Owren and Denise Tumiotto},
journal= {arXiv preprint arXiv:2308.08261},
year = {2024}
}