B-stability of numerical integrators on Riemannian manifolds

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

Lie Group integrators for mechanical systems

Since they were introduced in the 1990s, Lie group integrators have become a method of choice in many application areas. These include multibody dynamics, shape analysis, data science, image registration and biophysical simulations. Two important classes of intrinsic Lie group integrators are the Runge--Kutta--Munthe--Kaas methods and the commutator free Lie group integrators. We give a short introduction to these classes of methods. The Hamiltonian framework is attractive for many mechanical problems, and in particular we shall consider Lie group integrators for problems on cotangent bundles of Lie groups where a number of different formulations are possible. There is a natural symplectic structure on such manifolds and through variational principles one may derive symplectic Lie group integrators. We also consider the practical aspects of the implementation of Lie group integrators, such as adaptive time stepping. The theory is illustrated by applying the methods to two nontrivial applications in mechanics. One is the N-fold spherical pendulum where we introduce the restriction of the adjoint action of the group $SE(3)$ to $TS^2$, the tangent bundle of the two-dimensional sphere. Finally, we show how Lie group integrators can be applied to model the controlled path of a payload being transported by two rotors. This problem is modeled on $\mathbb{R}^6\times \left(SO(3)\times \mathfrak{so}(3)\right)^2\times (TS^2)^2$ and put in a format where Lie group integrators can be applied.Comment: 35 page