A Comparative Study of Projected and Unprojected Schemes for Micromagnetic Simulations

TL;DR

This study compares projected and unprojected semi-implicit schemes for micromagnetic simulations, analyzing their accuracy and stability with different dissipation coefficients.

Contribution

It provides a comparative analysis of two semi-implicit schemes with and without projection steps, highlighting their performance under various dissipation conditions.

Findings

Gauss-Seidel without projection differs significantly at high dissipation.

Both schemes can simulate steady states effectively with appropriate dissipation.

BDF1 scheme results are consistent regardless of projection, effectively modeling domain wall motion.

Abstract

In micromagnetic simulations, the constant magnitude of the magnetization can be derived from the continuity equation. Since the time evolution of the magnetization in the continuity equation is perpendicular to the plane determined by the magnetization and the effective field, taking the inner product of both sides of the model with the magnetization shows that the evolution rate of the magnitude of the magnetization is zero, thus keeping the magnitude constant. From this perspective, the equation itself can maintain the constraint of constant magnetization magnitude. We discretized the continuity equation and compared two first-order semi-implicit strategies in time: one is the implicit Gauss-Seidel method, and the other is the semi-implicit Backward Differentiation Formula (BDF) method. We considered the comparison between these two schemes with and without the projection step. The results of micromagnetic simulations show that when the dissipation coefficient is large, the implicit Gauss-Seidel method without the projection step has significant differences from the method with the projection step in both the achieved steady state and domain wall motion. When an appropriate dissipation coefficient is selected, the difference between the two narrows, and both the steady state and domain wall motion can be simulated. For the other method, BDF1, whether the dissipation coefficient is large or small, the results with and without the projection step are quite consistent, and it can effectively simulate the domain wall motion.