Improved Energy Stable Symmetric Gauss-Seidel Projection Method for Micromagnetics Simulations

Technical Summary: Improved Energy Stable Symmetric Gauss-Seidel Projection Method for Micromagnetics Simulations

Problem Statement
Micromagnetic simulations, governed by the Landau–Lifshitz–Gilbert (LLG) equation, are essential for studying magnetization dynamics in ferromagnetic materials. These simulations face significant numerical challenges due to the equation's stiffness, nonlinearity, and the strict unit-length constraint on magnetization vectors. While the Gauss–Seidel Projection Method (GSPM) has emerged as a robust, semi-implicit framework offering unconditional stability and computational efficiency, it possesses inherent limitations. Specifically, the standard GSPM relies on a single-sided (unidirectional) Gauss–Seidel iteration. This approach introduces directional numerical bias and can lead to degraded energy stability, particularly in systems with ultra-low Gilbert damping parameters ($\alpha \approx 10^{-3} \sim 10^{-4}$), which are common in high-quality magnetic materials like Yttrium Iron Garnet (YIG) and epitaxial FeCo alloys. In undamped ($\alpha=0$) or weakly damped regimes, the standard GSPM may fail to guarantee discrete energy stability, resulting in unphysical energy growth or oscillations.

Methodology
The authors propose a Symmetric Gauss–Seidel Projection Method (SGSPM) to address the stability deficiencies of the standard GSPM. The methodology retains the core structure of the GSPM—integrating fractional-step decomposition, manifold projection, and Fast Fourier Transform (FFT) techniques for solving linear systems—but modifies the iterative solver.

1. Symmetric Iteration: Unlike the standard GSPM, which performs a single forward Gauss–Seidel sweep ($1 \to n$), the SGSPM employs a two-pass symmetric iteration. This involves a forward sweep followed immediately by a backward sweep ($n \to 1$).
2. Algorithmic Steps:
   • Step 1 (Implicit Symmetric Gauss–Seidel): The magnetization field is updated using a symmetric iterative process that fully exploits the latest updated state variables from the heat-diffusion subproblem in both forward and backward directions.
   • Step 2 (Heat Flow without Constraints): An unconstrained heat flow step is performed to handle the damping and effective field terms.
   • Step 3 (Projection): A manifold projection step enforces the unit-length constraint ($|m|=1$) on the magnetization vector.
3. Theoretical Basis: The paper establishes that the symmetric iteration matrix $G_{SGS}$ satisfies $\rho(G_{SGS}) \leq \rho(G_{GS}) < 1$ for strictly diagonally dominant matrices, theoretically ensuring superior convergence and stability properties compared to the standard unidirectional scheme.

Key Contributions

• Development of SGSPM: The primary contribution is the formulation of the SGSPM, a variant that maintains the first-order temporal and second-order spatial accuracy of the original GSPM while significantly enhancing stability.
• Discrete Energy Stability: The method rigorously guarantees discrete energy stability. The authors demonstrate that the symmetric iteration eliminates the directional bias inherent in standard Gauss–Seidel sweeps, which is the root cause of energy instability in weakly damped systems.
• Robustness in Low-Damping Regimes: The scheme is specifically designed to handle ultra-low damping parameters ($\alpha \to 0$), a regime where standard explicit or semi-implicit methods often fail or require prohibitively small time steps.

Numerical Results
The performance of SGSPM was validated through extensive numerical experiments involving magnetization dynamic evolution and domain-wall propagation in both 1D and 3D settings.

• Energy Stability: In simulations with zero damping ($\alpha=0$), the standard GSPM exhibited unphysical energy growth and oscillations. In contrast, the SGSPM demonstrated strictly monotonic energy decay, adhering to physical energy dissipation laws.
• Damping Sensitivity: For positive damping values ($\alpha > 0$), both methods produced dissipative energy curves. However, SGSPM consistently achieved faster energy decay rates and lower residual energy levels compared to GSPM. As damping increased, the performance gap narrowed, but SGSPM remained superior.
• Accuracy and Efficiency: Numerical tests confirmed that SGSPM retains first-order temporal accuracy and second-order spatial accuracy. While the computational cost of SGSPM is slightly higher than GSPM due to the additional backward sweep, the use of FFT for solving the underlying linear systems keeps the overall efficiency high.
• Domain Wall Dynamics: Simulations of Néel domain wall motion under external fields showed that SGSPM produces numerically consistent and visually matching magnetization solutions compared to GSPM, confirming that the stability improvements do not compromise the physical fidelity of the dynamics.

Significance and Claims
The paper claims that the SGSPM offers a more robust numerical framework for micromagnetic simulations, particularly for modern applications involving ultra-low damping materials. By adopting a symmetric Gauss–Seidel strategy, the method effectively suppresses residual oscillations and directional numerical bias, ensuring that the discrete energy evolution remains physically valid even in the absence of Gilbert damping. The authors posit that this improvement makes SGSPM a superior choice for long-time dynamic simulations and high-frequency spin wave problems where energy stability is critical. Furthermore, the authors suggest that this symmetric strategy can be extended to higher-order GSPM variants and other improved schemes to further enhance stability under weak damping conditions.