Source-linked AI summary
Stability and error analysis of fully discrete original energy-dissipative and length-preserving scheme for the Landau-Lifshitz-Gilbert equation
Binghong Li, Xiaoli Li, Cheng Wang, Jiang Yang
TL;DR
The paper tackles the difficulty of combining practical projection-based discretization with point-wise length preservation, original energy dissipation, and rigorous error analysis for the LLG equation. It develops a linear fully discrete finite difference scheme and reformulates it weakly to control nonlinear terms. The resulting method preserves the stated structural properties and achieves theoretically optimal convergence rates, with numerical results showing second-order temporal and first-order spatial convergence.
Problem
Projection methods for the LLG equation are practical but may lose original energy dissipation, while fully discrete extensions and rigorous error analyses remain challenging.
Method
The paper constructs a linear fully discrete finite difference projection scheme and analyzes it through an equivalent weak reformulation exploiting point-wise length preservation.
Results
The scheme simultaneously preserves point-wise length, unconditional original energy dissipation, and stability, while the analysis establishes an optimal convergence rate; experiments show second-order temporal and first-order spatial convergence.
Takeaways & Limitations
The work provides a linear projection-based algorithm with the combined structural properties and convergence justification sought for fully discrete LLG computation.
Takeaways & Limitations
The analysis is subject to specific grid-ratio conditions, and extending the approach to mass-lumped finite elements is identified as future work.
Abstract
from arXiv · showhide
The Landau-Lifshitz-Gilbert (LLG) equation, regarded as a gradient flow with manifold constraint, is the fundamental model describing magnetization dynamics in ferromagnetic materials. It is well known that the normalized tangent plane method is able to simultaneously achieve the non-convex manifold constraint and original energy dissipation. However, the associated computational cost of this numerical approach is exceedingly high. By contrast, the projection method is more straightforward to implement, while it often compromises the inherent energy dissipative property of the continuous model, and the error analysis turns out to be even more challenging. In this work, we first construct a linear and fully discrete finite difference numerical scheme, based on the projection method for the LLG equation, which is capable of simultaneously preserving the non-convex manifold constraint \(|\mathbf{m}| = 1\) and an unconditional original energy dissipation. In the error analysis, the classical theoretical technique becomes ineffective, due to the presence of the nonlinear Laplacian term, which in turn poses a significant challenge. To overcome this subtle difficulty, we carefully rewrite the numerical method in an equivalent weak form, in which a point-wise length preserving feature of the numerical solution plays an essential role. As a result of these estimates in the reformulated weak form, an optimal convergence rate could be theoretically established. In our knowledge, this numerical method is the first linear algorithm that preserves the following combined theoretical properties: (i) point-wise length preservation, (ii) unconditional original energy dissipation, (iii) a theoretical justification of convergence analysis and optimal rate error estimate.
1. Introduction.
The paper addresses the challenge of designing a practical fully discrete LLG scheme that preserves the unit-length constraint, original energy dissipation, stability, and rigorous convergence. It proposes a linear projection-based finite difference method and an equivalent weak reformulation enabling optimal error estimates.
- Problem setting: The paper studies the LLG equation as a constrained magnetization model whose solution preserves point-wise magnitude and satisfies an energy dissipation law.The considered setting includes bounded domains in dimensions d ∈ {1, 2, 3}, with periodic or homogeneous Neumann boundary conditions.
- Motivation and related work: Projection-based schemes are straightforward but may compromise original energy dissipation, while prior analyses impose restrictive timestep–mesh conditions or lack the desired stability.The paper motivates preserving energy dissipation because it is physically important and supports convergence to local energy minima in skyrmion relaxation.
- Motivation and related work: Fully discrete extension of the semi-implicit projection approach and rigorous error analysis remain theoretically significant challenges.
- Main contributions: The paper constructs a linear, fully discrete finite difference projection scheme that enforces |m| = 1, unconditionally dissipates the original energy, and remains stable in the discrete H1 norm.The scheme is presented as straightforward to implement while preserving these properties simultaneously.
- Main contributions: An equivalent weak reformulation uses point-wise length preservation, discrete summation by parts, and interpolation–gradient relations to handle nonlinear terms in the error analysis.This reformulation avoids the highly complicated nonlinear term that obstructs direct estimates.
- Main contributions: The method is claimed to be the first linear LLG scheme combining point-wise length preservation, unconditional original energy dissipation, and theoretically justified optimal-rate error estimates.
2. Some preliminaries.
The preliminaries define the finite-difference grid, discrete operators, norms, boundary conditions, and inequalities used to analyze the scheme. They also introduce an equivalent reformulation that facilitates the fully discrete energy analysis.
- The analysis uses a uniform finite-difference grid with periodic boundary conditions, while homogeneous Neumann conditions admit a straightforward extension.
- Discrete gradient, central-difference Laplacian, inner-product, and norm operators are defined for grid functions.
- Summation by parts, inverse and interpolation inequalities, discrete Sobolev embedding, and a discrete Gronwall lemma provide the main analytical tools.
- Matched central interpolation satisfies discrete product identities for scalar, dot-product, and cross-product gradients.
- The equivalent formulation is introduced before the fully discrete scheme to facilitate theoretical analysis and derive unconditional original energy dissipation.
3. Semi-implicit projection scheme.
The projection-based finite-difference scheme is linear, uniquely solvable, point-wise length preserving, and unconditionally dissipative with respect to the original energy. Its error analysis uses an equivalent weak reformulation to control nonlinear terms and establish stability estimates.
- The fully discrete scheme preserves point-wise length |m^n| = 1 and the original energy dissipation.
- The intermediate stage forms a non-constant-coefficient linear system, and the fully discrete numerical system has a unique solution given m^(n−1).
- Orthogonality of the update terms yields |e m^n|^2 = |m^(n−1)|^2 + ∆t^2|m^(n−1) × (β∆_h e m^n + γ(m^(n−1) × ∆_h e m^n))|^2 ≥ 1.
- The original energy decreases unconditionally according to 1/2∥∇_h m^n∥^2_2 − 1/2∥∇_h m^(n−1)∥^2_2 ≤ −γ∆t∥m^(n−1) × ∆_h e m^n∥^2_2 ≤ 0.
- The error analysis rewrites the scheme in an equivalent weak form because the nonlinear Laplacian term makes direct estimates difficult.
4. Error estimate.
The error analysis reformulates the projection scheme in an equivalent weak form, using point-wise length preservation and discrete identities to control nonlinear terms and establish optimal convergence under grid-ratio conditions.
- Weak reformulation: The equivalent weak reformulation is derived from point-wise length preservation, discrete gradient identities, interpolation relations, and discrete summation by parts.This reformulation avoids highly complicated nonlinear terms in the original error equation.
- Error estimate: A sharper point-wise projected-error estimate follows from the constructed scheme’s lower bound |m~^n| ≥ 1, improving control over higher-order nonlinear terms.The estimate relates the intermediate and projected errors pointwise and is used in the convergence argument.
- Proof process: The proof also establishes unique solvability of the numerical system and uses discrete energy estimates, orthogonality, Hölder inequalities, and a priori bounds in the error argument.These ingredients support the stepwise control needed to recover the a priori assumptions and complete the optimal convergence proof.
- Error estimate: Theorem 4.3 establishes an optimal-rate error estimate for the fully discrete scheme when the exact solution has the stated regularity and the mesh parameters satisfy the required scaling.The result applies in 2D under h^2 ≲ ∆t ≲ h^ϵ0 and in 3D under h^2 ≲ ∆t ≲ h^(1+ϵ0), with sufficiently small h and ∆t.
- Energy structure: The reformulation connects the scheme derived from the original LLG formulation with the nonlinear |∇m|^2m term in an equivalent PDE formulation, supporting original energy-dissipation analysis.Projection-style schemes based directly on the alternative formulation generally face difficulty proving global-in-time energy dissipation.
5. Numerical results.
Numerical tests verify the scheme’s convergence, unconditional original-energy dissipation, blowup behavior, and static skyrmion configurations.
- 5.1. Convergence rate.: Second-order temporal and first-order spatial convergence rates agree with the theoretical predictions in discrete L2 and H1 norms.The tests use ∆t = h2 for temporal convergence and ∆t = h for spatial convergence.
- 5.2. Energy dissipation.: The original energy decreases monotonically for Gilbert damping parameters γ = 0.1, 0.5, 1, and 10, confirming unconditional dissipativity in the test.The simulation uses T = 1, β = 1, ∆t = 0.01, and a 100 × 100 grid on [0, 2π]2.
- 5.3. Phenomenon of blowup.: During the blowup test, mn preserves (0, 0, 1)T at the origin and turns toward (0, 0, −1)T nearby, consistent with reported blowup behavior.The evolution is shown at t = 0, 0.06, 0.15, 0.30, 0.32, and 0.35.
- 5.4. Static skyrmions.: The computed Q = 1 and Q = 0 static skyrmions match prior computational findings and experimental images for ϕ = π/2.The results include three-dimensional views, local state structures, and projections onto the x1x2-plane.