A Gauss-Seidel projection method with the minimal number of updates for stray field in micromagnetic simulations

TL;DR

This paper introduces GSPM-BDF2, a new semi-implicit scheme for micromagnetic simulations that significantly reduces computational time by updating the stray field only once per step, combining stability and efficiency.

Contribution

The paper proposes GSPM-BDF2, a novel method that merges GSPM and BDF2 advantages, achieving higher efficiency with minimal stray field updates in micromagnetic simulations.

Findings

GSPM-BDF2 reduces computational time by about 60% compared to GSPM.

It achieves 82-96% faster results than OOMMF on standard problems.

The method maintains stability and accuracy in simulations.

Abstract

Magnetization dynamics in magnetic materials is often modeled by the Landau-Lifshitz equation, which is solved numerically in general. In micromagnetic simulations, the computational cost relies heavily on the time-marching scheme and the evaluation of stray field. Explicit marching schemes are efficient but suffer from severe stability constraints, while nonlinear systems of equations have to be solved in implicit schemes though they are unconditionally stable. A better compromise between stability and efficiency is the semi-implicit scheme, such as the Gauss-Seidel projection method (GSPM) and the second-order backward differentiation formula scheme (BDF2). At each marching step, GSPM solves several linear systems of equations with constant coefficients and updates the stray field several times, while BDF2 updates the stray field only once but solves a larger linear system of equations with variable coefficients and a nonsymmetric structure. In this work, we propose a new method, dubbed as GSPM-BDF2, by combing the advantages of both GSPM and BDF2. Like GSPM, this method is first-order accurate in time and second-order accurate in space, and is unconditionally stable with respect to the damping parameter. However, GSPM-BDF2 updates the stray field only once per time step, leading to an efficiency improvement of about $60\%$ than the state-of-the-art GSPM for micromagnetic simulations. For Standard Problem \#4 and \#5 from National Institute of Standards and Technology, GSPM-BDF2 reduces the computational time over the popular software OOMMF by $82\%$ and $96\%$, respectively. Thus, the proposed method provides a more efficient choice for micromagnetic simulations.