Temporal upscaling in micro magnetism via heterogeneous multiscale methods
For researchers in micro-magnetism, this work provides a multiscale framework to overcome computational challenges from disparate temporal scales, though it is an incremental application of existing heterogeneous multiscale methods.
This paper introduces a heterogeneous multiscale method for the Landau-Lifschitz equations in micro-magnetism, enabling efficient simulation of magnetization dynamics without fully resolving microscopic scales. Numerical results for a single particle and a chain of interacting particles show accurate effective quantities with step-sizes much larger than microscopic scales.
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.