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

Yan Gui (School of Mathematical Sciences, Soochow University, Suzhou, China)

Cheng Wang (Mathematics Department, University of Massachusetts, North Dartmouth, MA, USA)

Jingrun Chen (School of Mathematical Sciences, University of Science and Technology of China, Hefei, China; and)

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

Received 13 November 2022

Received revised 19 February 2023

Accepted 18 December 2023

Published 15 July 2024