Geometric integration on spheres and some interesting applications
For researchers in numerical analysis and computational physics, this work provides novel geometric integrators for constrained ODEs/PDEs on spheres, but the results are incremental as they extend existing geometric integration theory.
The paper develops one-step geometric integration algorithms for ODEs on spheres and demonstrates their application to rigid body motion and micromagnetics, linking them to differential geometry via partial connection forms.
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.