Canonical structure of the LLG equation for exponential updates in micromagnetism

Abstract

In this contribution, we propose an exponential update algorithm for magnetic moments appearing in the framework of micromagnetics and the Landau–Lifshitz–Gilbert (LLG) equation. This algorithm can be interpreted as the geometric integration on spheres that a priori satisfies the unit length constraint of the normalized magnetization vector. Even though the geometric structures for this are obvious and some works already use an exponential algorithm, to the best of the authors’ knowledge, there is no canonical structure of the LLG equation for the exponential update algorithm in micromagnetism. Tensor algebraic reformulations of the LLG equation allow the canonical representation of the evolution equation for magnetization, which serves as the basis for different integrators. Based on the specific structure of the exponential of skew symmetric matrices, an efficient update scheme is derived. The excellent performance of the proposed exponential update algorithm is demonstrated in representative examples.