Geometric integration on spheres and some interesting applications

TL;DR

This paper introduces new one-step geometric integrators for solving ODEs on spheres, demonstrating their effectiveness in modeling rigid body dynamics and micromagnetics, and linking them to differential geometry concepts.

Contribution

It presents novel algorithms for geometric integration on spheres, connecting numerical schemes with differential geometry via partial connection forms.

Findings

Effective algorithms for ODEs on spheres demonstrated

Applications to rigid body motion and micromagnetics shown

Theoretical framework links geometric integration to differential geometry

Abstract

Geometric integration theory can be employed when numerically solving ODEs or PDEs with constraints. In this paper, we present several one-step algorithms of various orders for ODEs on a collection of spheres. To demonstrate the versatility of these algorithms, we present representative calculations for reduced free rigid body motion (a conservative ODE) and a discretization of micromagnetics (a dissipative PDE). We emphasize the role of isotropy in geometric integration and link numerical integration schemes to modern differential geometry through the use of partial connection forms; this theoretical framework generalizes moving frames and connections on principal bundles to manifolds with nonfree actions.