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Efficient Hermitian and skew-Hermitian splitting methods for linear systems in micromagnetic simulations

This paper proposes and evaluates Hermitian/skew-Hermitian splitting (HSS) and inexact HSS (IHSS) methods as efficient solvers for the large-sparse, non-Hermitian positive-definite linear systems arising from semi-implicit discretizations of the Landau-Lifshitz equation in micromagnetic simulations, demonstrating their convergence and parameter sensitivity through numerical experiments.

Original authors: Yingxi Miao, Changjian Xie

Published 2026-08-26

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CC BY 4.0

Original authors: Yingxi Miao, Changjian Xie

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of hard drives, electric motors, and the emerging field of spintronics, tiny magnetic particles act as the fundamental building blocks of modern technology. These particles are not static; their internal magnetic direction, known as magnetization, constantly shifts and rotates in response to external fields and internal forces. To predict how these materials behave, scientists rely on a mathematical description called the Landau-Lifshitz equation. This equation captures two competing physical behaviors: a spinning motion where the magnetic direction precesses around an effective field, and a damping effect that gradually slows this spin until it settles. A critical rule governs this entire process: the strength of the magnetization must remain constant, like a spinning top that never changes its height, only its tilt. Because the equations governing this motion are complex and change over time, scientists cannot solve them with simple pen-and-paper math. Instead, they must break the problem down into millions of tiny steps, creating a massive grid of numbers that a computer can crunch.

The challenge lies in the sheer size and nature of these number grids. As researchers try to simulate more realistic materials with finer detail, the number of equations grows to the millions. These systems are "sparse," meaning most of the numbers in the grid are zero, but they are also "non-Hermitian," a technical way of saying the numbers do not follow a simple, symmetrical pattern that makes them easy to solve. Traditional computer methods for solving such systems often struggle, either running out of memory or getting stuck in a loop where the answer never settles down. This creates a bottleneck for designing better magnetic devices, as engineers need fast, reliable simulations to test new ideas before building them.

In this work, researchers Yingxi Miao and Changjian Xie tackled this computational bottleneck by applying a specific strategy known as Hermitian/skew-Hermitian splitting. Imagine trying to solve a complex puzzle where the pieces are locked in two different ways: some pieces fit together in a perfectly symmetrical, predictable manner, while others are twisted and asymmetrical. Instead of trying to force all the pieces together at once, this method separates the puzzle into its symmetrical and asymmetrical parts. The researchers then solve these two parts in alternating steps, using the solution from one step to help solve the next. They also developed a "rough" version of this method, called inexact splitting, which allows the computer to take a slightly less precise path through the symmetrical part of the puzzle to save time, provided the final result is still accurate enough.

The team tested this approach on two different ways of writing down the magnetic equations, both of which are third-order semi-implicit schemes. These are advanced time-stepping methods that allow for larger, faster steps in the simulation while maintaining stability. By applying their splitting technique to these schemes, they created a solver that could handle the massive, irregular grids generated by the simulations. The researchers ran extensive tests in both one-dimensional and three-dimensional models, simulating magnetic materials with different levels of damping and varying grid sizes. They measured how quickly the computer iterations converged to a solution and compared the actual speed of convergence against the theoretical limits predicted by math.

The results showed that the method worked exactly as the theory suggested. In every test case, the computer iterations moved steadily toward the correct answer without stalling or diverging. The speed of this convergence was consistently faster than the worst-case scenario predicted by the mathematical formulas, meaning the method is robust and reliable. The researchers found that the theoretical upper limit for the convergence speed was a very accurate predictor of what would happen in practice. Furthermore, the "rough" or inexact version of the method proved to be highly effective, provided the parameters were chosen carefully. While the specific choice of a splitting parameter influenced the efficiency, the method remained stable and fast across a wide range of conditions.

When comparing the two different schemes they tested, the researchers found that both performed similarly in terms of convergence behavior. However, they noted that one of the schemes, which treats the damping forces in a slightly different way, required more computational time to reach the same level of accuracy as the other. This suggests that while the underlying solver is powerful, the specific way the physical equations are written can still impact how hard the computer has to work. In three-dimensional simulations, the method maintained its stability, keeping the number of steps required to solve the problem relatively low even as the grid became more detailed. The final magnetic patterns produced by both schemes were nearly identical, confirming that the different mathematical approaches lead to the same physical reality.

Ultimately, this research provides a reliable tool for scientists and engineers working with magnetic materials. By proving that this splitting method can efficiently solve the difficult, large-scale equations that arise in micromagnetic simulations, the authors have removed a significant hurdle from the path of discovery. The method does not just work in theory; it performs consistently in complex, three-dimensional environments, offering a clear path forward for simulating the behavior of next-generation spintronic devices and magnetic storage media. The work confirms that by breaking a difficult problem into its symmetrical and asymmetrical components, even the most stubborn computational challenges can be solved with speed and precision.

Technical Summary: Efficient Hermitian and Skew-Hermitian Splitting Methods for Linear Systems in Micromagnetic Simulations

Problem Statement
The paper addresses the computational challenges inherent in simulating the Landau-Lifshitz-Gilbert (LLG) equation, which governs the time evolution of magnetization in ferromagnetic materials. Practical numerical simulations require solving large, sparse linear systems at each time step. When employing semi-implicit time discretization schemes (specifically third-order backward differentiation formulas combined with one-sided extrapolation), the resulting coefficient matrices are non-Hermitian positive definite (NHPD).

While the matrices possess positive real parts in their eigenvalues (ensuring invertibility), their non-Hermitian nature—arising from the coexistence of symmetric positive definite components and skew-symmetric components—poses significant difficulties for traditional iterative solvers. Classical Krylov subspace methods (e.g., GMRES, BiCGStab) often suffer from high memory consumption, residual oscillation, or slow convergence for such systems. Conversely, standard splitting methods designed for Hermitian systems fail to maintain stability for NHPD matrices.

Methodology
To overcome these limitations, the authors apply the Hermitian/Skew-Hermitian Splitting (HSS) method and its inexact variant (IHSS) to the linear systems generated by two distinct third-order semi-implicit schemes (Scheme A and Scheme B) for the LLG equation.

Matrix Decomposition: The core of the approach involves decomposing the coefficient matrix $A$ (or $B$ ) into its Hermitian part $H = \frac{1}{2}(A + A^*)$ and skew-Hermitian part $S = \frac{1}{2}(A - A^*)$ . For the NHPD systems derived from the LLG equation, $H$ is strictly positive definite.
HSS Iteration: The method constructs an alternating iteration scheme that separately processes the positive definite Hermitian component and the skew-Hermitian component. Theoretical analysis confirms that this iteration converges unconditionally for any positive relaxation parameter $\beta$ .
Inexact HSS (IHSS): Given the large-scale sparse nature of the problems, exact solutions to the sub-systems in each HSS step are computationally expensive. The authors employ the IHSS method, which replaces exact direct solves with inner iterative solvers (specifically Preconditioned Conjugate Gradient and Aggregation-based Algebraic MultiGrid). The inner iterations are terminated based on a residual tolerance relative to the outer iteration residual.
Parameter Optimization: The study investigates the optimal splitting parameter $\beta^*$ , theoretically derived as the geometric mean of the extreme eigenvalues of $H$ , to minimize the spectral radius of the iteration matrix.

Key Contributions

Application to Micromagnetics: The paper extends the HSS/IHSS framework specifically to the variable-coefficient linear systems arising from high-order semi-implicit discretizations of the LLG equation in both 1D and 3D domains.
Theoretical Validation: The authors provide a convergence analysis showing that the spectral radius of the HSS iteration matrix is strictly bounded by a value less than one, ensuring unconditional convergence.
Comparative Analysis of Schemes: The study compares two specific semi-implicit formulations (Scheme A and Scheme B). While both yield NHPD systems, the authors analyze their structural differences and the resulting performance of the linear solvers.
Parameter Sensitivity: The work demonstrates that the efficiency of the IHSS method is highly sensitive to the choice of the splitting parameter $\beta$ , showing that theoretically derived optimal parameters significantly reduce iteration counts compared to arbitrary choices.

Numerical Results
Extensive numerical experiments were conducted to validate the theoretical findings:

Spectral Radius Behavior: In both 1D and 3D simulations across various grid resolutions ( $N_x$ ) and damping parameters ( $\alpha$ ), the actual spectral radius of the iteration matrix remained strictly below its theoretical upper bound and below unity. The theoretical bound was found to be a sharp and accurate estimate of the actual convergence behavior.
IHSS Performance: The IHSS method demonstrated effective convergence for all tested cases. The average number of iterations was generally low, and relative residuals were reduced to approximately $10^{-7}$ .
Parameter Sensitivity: Results confirmed that the convergence speed is sensitive to $\beta$ . Using the optimal $\beta$ (e.g., $\approx 1.833$ ) drastically reduced the average iteration count compared to non-optimal choices (e.g., $\beta=1$ or $\beta=10$ ).
Scheme Comparison: While Scheme B required slightly more CPU time to achieve the same accuracy level as Scheme A (indicating a more challenging linear system), both schemes exhibited comparable convergence behavior when solved with the IHSS method. The final solution profiles for both schemes were nearly identical.

Significance and Claims
The paper claims that the HSS and IHSS methods provide a robust, efficient, and unconditionally convergent alternative to traditional solvers for the specific class of NHPD linear systems encountered in micromagnetic simulations. The study validates that:

The theoretical convergence bounds for HSS are tight and reliable for these physical models.
The IHSS method successfully handles the computational overhead of large sparse systems without sacrificing stability.
The proposed approach offers a viable pathway for high-performance numerical simulations of domain-wall motion and spin-wave propagation, where the preservation of the magnetization magnitude constraint and numerical stability are critical.

The authors conclude that while the proposed semi-implicit schemes generate more complex linear systems than lower-order counterparts, the application of HSS/IHSS methods effectively mitigates the associated computational challenges, making them suitable for practical engineering and scientific computing applications in micromagnetics.

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