Adaptively time stepping the stochastic Landau-Lifshitz-Gilbert equation at nonzero temperature: implementation and validation in MuMax3

TL;DR

This paper introduces an adaptive time stepping algorithm for the stochastic Landau-Lifshitz-Gilbert equation at nonzero temperature, implemented in MuMax3, resulting in faster simulations without sacrificing accuracy.

Contribution

It extends high order solvers with adaptive time stepping for stochastic micromagnetic simulations, validated in MuMax3, enabling more efficient and accurate modeling at nonzero temperatures.

Findings

Twenty-fold speedup over fixed time step methods

Validated accuracy of the adaptive algorithm

Implemented in GPU-accelerated MuMax3

Abstract

Thermal fluctuations play an increasingly important role in micromagnetic research relevant for various biomedical and other technological applications. Until now, it was deemed necessary to use a time stepping algorithm with a fixed time step in order to perform micromagnetic simulations at nonzero temperatures. However, Berkov and Gorn have shown that the drift term which generally appears when solving stochastic differential equations can only influence the length of the magnetization. This quantity is however fixed in the case of the stochastic Landau-Lifshitz-Gilbert equation. In this paper, we exploit this fact to straightforwardly extend existing high order solvers with an adaptive time stepping algorithm. We implemented the presented methods in the freely available GPU-accelerated micromagnetic software package MuMax3 and used it to extensively validate the presented methods. Next to the advantage of having control over the error tolerance, we report a twenty fold speedup without a loss of accuracy, when using the presented methods as compared to the hereto best practice of using Heun's solver with a small fixed time step.