Higher-Order Norm Preserving and Energy Decreasing IMEX Schemes for the Landau-Lifshitz Equation

Temel Kavramlar

The authors construct a new class of higher-order implicit-explicit (IMEX) schemes for the Landau-Lifshitz equation that preserve the pointwise length constraint, satisfy a modified energy dissipation law, and have rigorous error estimates.

Özet

The authors present a new class of higher-order IMEX schemes for the Landau-Lifshitz equation. The key features of these schemes are:

They are purely linear and only require solving decoupled or coupled elliptic equations at each time step.
They preserve the pointwise length constraint of the magnetization field.
They satisfy a modified energy dissipation law, and the numerical solutions are uniformly bounded.
The authors carry out a rigorous error analysis for the semi-discretized schemes up to fifth-order in a unified framework, establishing error estimates in l^∞(0, T; H^1(Ω)) ∩ l^2(0, T; H^2(Ω)) under mild conditions on the time step size and the exchange parameter.

The authors first reformulate the Landau-Lifshitz equation into an equivalent form that is more amenable to numerical treatment. They then construct the IMEX-GSAV (Generalized Scalar Auxiliary Variable) schemes and prove that the numerical solutions are uniformly bounded. Finally, they perform a detailed error analysis, deriving rigorous error estimates for the semi-discretized schemes up to fifth-order.

İstatistikler

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Alıntılar

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Önemli Bilgiler Şuradan Elde Edildi

On a class of higher-order length preserving and energy decreasing IMEX schemes for the Landau-Lifshitz equation

by Xiaoli Li,Na... : arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.08902.pdf

Daha Derin Sorular

How can the proposed IMEX-GSAV schemes be efficiently implemented with finite element methods, especially when the parameter β is relatively large

The proposed IMEX-GSAV schemes can be efficiently implemented with finite element methods, even when the parameter β is relatively large. When dealing with finite element methods, especially in the case where β is significant, a common approach is to combine the IMEX-GSAV schemes with Gauss-Seidel techniques. This combination allows for the treatment of the term βBl(mn) × ∆Bl(mn) semi-implicitly, which can be beneficial when β is large. By using Gauss-Seidel techniques, the implicit treatment of the nonlinear term can be handled effectively, ensuring stability and accuracy in the numerical solution. Additionally, the use of Gauss-Seidel methods can help in overcoming the challenges posed by the explicit treatment of the β term, especially in the context of finite element implementations.

Can the error estimates be extended to fully discretized schemes without severe time step constraints

The error estimates provided for the semi-discretized schemes can potentially be extended to fully discretized schemes without imposing severe time step constraints. The rigorous error analysis conducted for the IMEX-GSAV schemes up to fifth-order in a unified framework lays a strong foundation for extending these estimates to fully discretized schemes. By carefully considering the properties of the fully discretized schemes and leveraging the error estimates derived for the semi-discretized versions, it is possible to establish error bounds for the fully discretized schemes. This extension would provide valuable insights into the accuracy and stability of the numerical solutions obtained from the fully discretized IMEX-GSAV schemes.

Are there any other equivalent formulations of the Landau-Lifshitz equation that could lead to more efficient numerical schemes

There are several other equivalent formulations of the Landau-Lifshitz equation that could potentially lead to more efficient numerical schemes. One interesting formulation is the heat flow for harmonic maps, which is a reformulation of the Landau-Lifshitz equation. This formulation has been well studied and offers a different perspective on the dynamics of ferromagnetic materials. By exploring alternative formulations and their numerical implications, researchers can potentially discover new approaches to simulating magnetization dynamics with improved efficiency and accuracy. Additionally, variations in the formulation of the Landau-Lifshitz equation may offer insights into different aspects of the physics involved, leading to novel numerical schemes with specific advantages in certain scenarios.