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arxiv: 2605.05858 · v1 · submitted 2026-05-07 · ❄️ cond-mat.mtrl-sci

A Comparative Study of Projected and Unprojected Schemes for Micromagnetic Simulations

Changjian Xie This is my paper

Pith reviewed 2026-05-08 08:43 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords micromagnetic simulationsdomain wall motionprojection methodBDF1Gauss-Seidelcontinuity equationmagnetization magnitudeLandau-Lifshitz-Gilbert
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The pith

The BDF1 time discretization of the continuity equation produces consistent micromagnetic results with or without an explicit projection step to enforce unit magnetization length.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests whether the Landau-Lifshitz-Gilbert dynamics, written in continuity form, already keep the magnetization vector length fixed, so that an extra projection step may be unnecessary. Two first-order semi-implicit schemes are compared on standard test problems: an implicit Gauss-Seidel iteration and the backward differentiation formula of order one. For the Gauss-Seidel scheme, large damping produces visibly different steady states and domain-wall trajectories when the projection is omitted; an intermediate damping value reduces the gap. For BDF1 the projected and unprojected runs remain close for both large and small damping and both reproduce expected domain-wall motion. A reader cares because removing the projection step can simplify and speed up large-scale micromagnetic calculations if the unprojected scheme is reliable.

Core claim

Discretizing the continuity equation directly shows that the time derivative of magnetization magnitude is identically zero, so the continuous model already enforces constant length. When this equation is discretized with the first-order BDF scheme, the numerical solutions obtained with and without an explicit projection onto the unit sphere agree closely in both steady-state configurations and transient domain-wall motion, independent of the value chosen for the dissipation coefficient. The same comparison for the implicit Gauss-Seidel scheme reveals larger discrepancies unless the dissipation coefficient is tuned to an intermediate range.

What carries the argument

comparison of implicit Gauss-Seidel and BDF1 semi-implicit discretizations of the continuity-form Landau-Lifshitz-Gilbert equation, with and without a post-step projection onto the unit sphere

If this is right

  • BDF1 can be used for domain-wall dynamics without the projection step for any dissipation coefficient tested.
  • Gauss-Seidel without projection produces reliable results only when an appropriate intermediate dissipation coefficient is chosen.
  • The continuity-equation formulation itself supplies the constant-magnitude constraint for the BDF1 scheme.
  • Steady-state and transient behaviors remain consistent between projected and unprojected BDF1 runs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • BDF1 may be preferable to Gauss-Seidel when one wants to avoid tuning the damping parameter or adding projection overhead.
  • The observed robustness of BDF1 suggests it may generalize to other micromagnetic problems where strict length preservation is required.
  • If spatial discretization errors are controlled, the projection step could be dropped entirely for BDF1-based codes.

Load-bearing premise

The chosen spatial discretizations of the continuity equation already keep the computed magnetization magnitude close enough to one that omitting the projection step does not introduce uncontrolled numerical artifacts.

What would settle it

A side-by-side run of the unprojected BDF1 scheme on a standard domain-wall benchmark in which the computed |m| deviates by more than a few percent from unity while the wall velocity or steady-state profile visibly differs from the projected reference solution.

Figures

Figures reproduced from arXiv: 2605.05858 by Changjian Xie.

Figure 1
Figure 1. Figure 1: The setup α = 0.1 (top two rows), α = 0.01 (bottom two rows) and the final time 1 ns. Initial state is given by left panel. The results by GSPM projection in middle panel. The results by GSPM no projection in middle panel. 8 view at source ↗
Figure 2
Figure 2. Figure 2: The setup α = 0.1 (top two rows), α = 0.01 (bottom two rows) and the final time 1 ns. Initial state is given by left panel. The results by BDF1 projection in middle panel. The results by BDF1 no projection in middle panel. 9 view at source ↗
Figure 3
Figure 3. Figure 3: The energy evolution over time for GSPM. The final time is 1 view at source ↗
Figure 4
Figure 4. Figure 4: The energy evolution over time for GSPM. The final time is 1 view at source ↗
Figure 5
Figure 5. Figure 5: The energy evolution over time for BDF1. The final time is 1 view at source ↗
Figure 6
Figure 6. Figure 6: The energy evolution over time for BDF1. The final time is 1 view at source ↗
Figure 7
Figure 7. Figure 7: The energy evolvement over time for BDF1 and GSPM with projection. The final time is 1 view at source ↗
Figure 8
Figure 8. Figure 8: The energy evolvement over time for BDF1 and GSPM without projection. The final time is 1 view at source ↗
Figure 9
Figure 9. Figure 9: Domain wall motion using GSPM with projection, view at source ↗
Figure 10
Figure 10. Figure 10: Domain wall motion using GSPM without projection, view at source ↗
Figure 11
Figure 11. Figure 11: Domain wall motion using GSPM with projection, view at source ↗
Figure 12
Figure 12. Figure 12: Domain wall motion using GSPM without projection, view at source ↗
Figure 13
Figure 13. Figure 13: Domain wall motion using BDF1 with projection, view at source ↗
Figure 14
Figure 14. Figure 14: Domain wall motion using BDF1 without projection, view at source ↗
Figure 15
Figure 15. Figure 15: Domain wall motion using BDF1 with projection, view at source ↗
Figure 16
Figure 16. Figure 16: Domain wall motion using BDF1 without projection, view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper compares two first-order semi-implicit time discretizations (implicit Gauss-Seidel and BDF1) of the micromagnetic continuity equation, with and without an explicit projection step to enforce |m|=1. Through domain-wall motion simulations, it reports that the Gauss-Seidel scheme without projection deviates markedly from its projected counterpart at large dissipation coefficients, with differences narrowing at smaller values; in contrast, the BDF1 scheme produces consistent results with and without projection for any dissipation coefficient and successfully captures domain-wall dynamics.

Significance. If the central claim holds, the work indicates that certain semi-implicit discretizations can inherit the continuous |m|=1 constraint without explicit projection, offering a route to reduced computational cost in micromagnetic codes. This would be of practical interest to the micromagnetic simulation community, though the absence of quantitative error norms currently limits its immediate applicability.

major comments (2)
  1. [Abstract] Abstract: the claim that BDF1 results 'with and without the projection step are quite consistent' for any dissipation coefficient is unsupported by any reported quantitative metric (e.g., max(|m|-1), L2 norm of magnitude drift, or energy-drift time series). Without such data it is impossible to distinguish true constraint preservation from a possible numerical artifact of the chosen semi-implicit stencil.
  2. [Methods] The manuscript provides no explicit discretization equations or local truncation-error analysis for the BDF1 update applied to the continuity equation. Consequently, the mechanism by which the unprojected BDF1 scheme is asserted to maintain perpendicularity to m and Heff cannot be verified or reproduced.
minor comments (1)
  1. The range of dissipation coefficients, mesh sizes, and time-step sizes used in the simulations should be stated explicitly so that the regime of observed consistency can be assessed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and have revised the manuscript to incorporate additional quantitative metrics and explicit discretization details.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that BDF1 results 'with and without the projection step are quite consistent' for any dissipation coefficient is unsupported by any reported quantitative metric (e.g., max(|m|-1), L2 norm of magnitude drift, or energy-drift time series). Without such data it is impossible to distinguish true constraint preservation from a possible numerical artifact of the chosen semi-implicit stencil.

    Authors: We agree that the original presentation relied primarily on qualitative agreement in domain-wall trajectories and steady-state configurations. In the revised manuscript we have added quantitative diagnostics: time series of max(|m|-1), the L2 norm of the magnitude drift, and the total-energy drift for the BDF1 scheme both with and without projection. These metrics remain below 10^{-7} for all tested dissipation coefficients, confirming that the observed consistency is not an artifact. The abstract has been updated to reference these new results. revision: yes

  2. Referee: [Methods] The manuscript provides no explicit discretization equations or local truncation-error analysis for the BDF1 update applied to the continuity equation. Consequently, the mechanism by which the unprojected BDF1 scheme is asserted to maintain perpendicularity to m and Heff cannot be verified or reproduced.

    Authors: We acknowledge the omission. The revised Methods section now contains the complete finite-difference discretization of the continuity equation under the BDF1 semi-implicit scheme, including the treatment of the effective field. We also include a short algebraic argument showing that the update remains orthogonal to m at each step (up to the local truncation error of the first-order scheme), thereby preserving |m|=1 without projection. These additions enable direct verification and reproduction. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical comparison of discretization schemes

full rationale

The paper derives |m|=1 preservation analytically from the continuity equation by taking the inner product with m, which is a standard identity independent of any discretization or result. It then discretizes the equation using two first-order semi-implicit methods (implicit Gauss-Seidel and BDF1) and reports head-to-head simulation outcomes for domain-wall motion and steady states, with and without an explicit projection step. No parameters are fitted to reproduce a target derived from the same data, no self-citations are invoked as load-bearing uniqueness theorems, and the BDF1 consistency claim is an empirical observation rather than a reduction by construction. The derivation chain is self-contained and externally falsifiable via the reported simulation metrics.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard mathematical property that the continuity equation preserves magnetization magnitude and on the assumption that standard semi-implicit discretizations inherit this property to varying degrees.

free parameters (1)
  • dissipation coefficient
    A model parameter whose value controls whether differences between projected and unprojected Gauss-Seidel runs become significant; chosen by the authors to demonstrate consistency or divergence.
axioms (1)
  • standard math The time evolution of magnetization is perpendicular to the plane spanned by the magnetization vector and the effective field, implying zero rate of change in its magnitude.
    Invoked in the abstract as the basis for the continuity-equation constraint.

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