Temporal upscaling in micro magnetism via heterogeneous multiscale methods

TL;DR

This paper introduces a multiscale computational approach for simulating micro-magnetic dynamics efficiently by coupling microscopic and macroscopic models, enabling accurate macro-scale simulations without fully resolving microscopic details.

Contribution

It develops a heterogeneous multiscale method for Landau-Lifschitz equations, allowing macro-model parameters to be computed dynamically, improving efficiency over traditional schemes.

Findings

Effective quantities can be computed with larger step-sizes.

The method accurately captures single spin dynamics under high frequency fields.

Numerical validation on magnetic particles confirms theoretical predictions.

Abstract

We consider a multiscale strategy addressing the disparate scales in the Landau-Lifschitz equations in micro-magnetism. At the microscopic scale, the dynamics of magnetic moments are driven by a high frequency field. On the macroscopic scale we are interested in simulating the dynamics of the magnetisation without fully resolving the microscopic scales. The method follows the framework of heterogeneous multiscale methods and it has two main ingredients: a micro- and a macroscale model. The microscopic model is assumed to be known exactly whereas the macro model is incomplete as it lacks effective quantities. The two models use different temporal and spatial scales and effective parameter values for the macro model are computed on the fly, allowing for improved efficiency over traditional one-scale schemes. For the analysis, we consider a single spin under a high frequency field and show that effective quantities can be obtained accurately with step-sizes much larger than the size of the microscopic scales required to resolve the microscopic features. Numerical results both for a single magnetic particle as well as a chain of interacting magnetic particles are given to validate the theory.