Geometric integration on spheres and some interesting applications

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.Comment: This paper appeared in prin

Lagrangian mechanics and variational integrators on two‐spheres

“…It is interesting to see that the reprojection approach makes the total energy error worse, even though it preserves the structure of (S 2 ) n accurately. This shows that a standard reprojection method can corrupt numerical trajectories [5,13].…”

“…If a group acts transitively on a manifold, a curve on the manifold can be represented as the action of a curve in the Lie group on an initial point on the manifold. As such, Lie group methods can be applied to obtain numerical integration schemes for homogeneous manifolds [19,13,14]. However, it is not guaranteed that these methods preserve the geometric properties of the dynamics.…”

“…Using the characteristics of the homogeneous manifold, the discrete update map is represented by a group action of SO(3), and a proper subspace is searched to obtain a compact, possibly explicit, form for the numerical integrator. As a result, the following numerical problems are avoided: (i) local parameterizations yield singularities; (ii) numerical trajectories in the vicinity of a singularity experiences numerical ill-conditioning; (iii) unit length constraints lead to additional computational complexity; (iv) reprojection corrupts the numerical accuracy of trajectories [5,13].…”

An electro‐elastic phase‐field model for nematic liquid crystal elastomers based on Landau‐de‐Gennes theory

“…This choice allows for a natural modeling of the characteristic half-integer defects and a straight-forward numerical implementation. Director models usually have to deal with the unity constraint on the nematic director, which is a numerical challenge also known from micro-magnetic theories (see, for example, E & WANG [24], LEWIS & NIGAM [35], ZHANG & CHEN [73,72] and MIEHE & ETHIRAJ [38]). …”

An Efficient Geometric Integrator for Thermostatted Anti-/Ferromagnetic Models

“…LL and LLG are currently a very active topic of research. Other approaches to them can be found in [9,23,6,10,19,21,14,17], who also provide further references. However none of these consider a thermostatted version.…”