B-stability of numerical integrators on Riemannian manifolds

TL;DR

This paper extends the concept of B-stability of numerical integrators to Riemannian manifolds, demonstrating stability for geodesic Implicit Euler and analyzing its properties through theoretical and numerical results.

Contribution

It introduces a novel notion of B-stability on Riemannian manifolds, including stability results for geodesic Implicit Euler and new error estimates for Lie group integrators.

Findings

GIE is B-stable on manifolds with non-positive curvature.

GIE can be expansive on the 2-sphere with certain vector fields.

Large step sizes may lead to non-uniqueness of solutions.

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.