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Communications in Mathematical Sciences
Volume 22 (2024)
Number 5
Implicit-explicit Runge-Kutta methods for Landau-Lifshitz equation with arbitrary damping
Pages: 1397 – 1425
DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n5.a9
Authors
Abstract
Magnetization dynamics in ferromagnetic materials is modeled by the Landau-Lifshitz (LL) equation, a nonlinear system of partial differential equations. Among the numerical approaches, semi-implicit schemes are widely used in the micromagnetics simulation, due to a nice compromise between accuracy and efficiency. At each time step, only a linear system needs to be solved and a projection is then applied to preserve the length of magnetization. However, this linear system contains variable coefficients and a non-symmetric structure, and thus an efficient linear solver is highly desired. If the damping parameter becomes large, it has been realized that efficient solvers are only available to a linear system with constant, symmetric, and positive definite (SPD) structure. In this work, based on the implicit-explicit Runge-Kutta (IMEX-RK) time discretization, we introduce an artificial damping term, which is treated implicitly. The remaining terms are treated explicitly. This strategy leads to a semi-implicit scheme with the following properties: (1) only a few linear systems with constant and SPD structure needs to be solved at each time step; (2) it works for the LL equation with arbitrary damping parameter; (3) high-order accuracy can be obtained with high-order IMEX-RK time discretization. Numerically, second-order and third-order IMEX-RK methods are designed in both the 1-D and 3-D domains. A comparison with the backward differentiation formula scheme is undertaken, in terms of accuracy and efficiency. The robustness of both numerical methods is tested on the first benchmark problem from National Institute of Standards and Technology. The linearized stability estimate and optimal rate convergence analysis are provided for an alternate IMEX-RK2 numerical scheme as well.
Keywords
micromagnetics simulation, Landau–Lifshitz equation, implicit-explicitly Runge–Kutta scheme, second-order accuracy, third-order accuracy, hysteresis loop
2010 Mathematics Subject Classification
35K61, 65N06, 65N12
This work is supported in part by the grants NSFC 11971021 (J. Chen), NSF DMS-2012269, and DMS 2309548 (C. Wang).
Received 13 November 2022
Received revised 19 February 2023
Accepted 18 December 2023
Published 15 July 2024