Cayley Transforms in Micromagnetics

Abstract

Methods of numerical integration of ordinary differential equations exploiting the Cayley transform arise in a variety of contexts, ranging from the classical mid-point rule to symplectic and (almost) Poisson integrators, to numerical methods on Lie Groups. In earlier work, the first author investigated the interplay between the Cayley transform and the Jacobi identity in establishing certain error formulas for the mid-point rule (with applications to coupled rigid bodies). In this paper, we use the Cayley transform to lift the Landau-Lifshitz-Gilbert equation of micromagnetics to the Lie algebra of the group of currents (on a compact magnetic body) with values in the 3-dimensional rotation group. This follows an idea of Arieh Iserles and, we use the lift to numerically integrate the Landau-Lifshitz-Gilbert equation conserving automatically the norm of the magnetization everywhere.