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.

翻译：我们提出将数值单步积分器的非线性稳定性推广至黎曼流形，该推广遵循Butcher提出的B稳定性概念。受Simpson-Porco与Bullo工作的启发，我们在该类流形上引入非扩张系统，并定义了积分器的B稳定性。在本次初步阐述中，我们针对测地线版本隐式欧拉（GIE）格式给出了具体结果。证明了在具有非正截面曲率的黎曼流形上，GIE方法具有B稳定性。通过数值算例表明，当将GIE方法应用于二维球面上的特定非扩张向量场时会出现扩张现象，且对于足够大的步长，该方法不一定存在唯一解。最后，我们推导出一般李群积分器新的改进全局误差估计。