A Third-order Implicit-Explicit Runge-Kutta Method for Landau-Lifshitz Equation with Arbitrary Damping Parameters

TL;DR

This paper introduces a third-order implicit-explicit Runge-Kutta scheme for the Landau-Lifshitz-Gilbert equation, offering improved efficiency, accuracy, and stability for simulating magnetization dynamics with arbitrary damping parameters.

Contribution

A novel third-order implicit-explicit Runge-Kutta method that reduces computational cost and is unconditionally stable for arbitrary damping parameters in magnetization modeling.

Findings

Linear system with constant coefficients at each stage reduces computational cost

Method achieves third-order accuracy in 1-D and 3-D simulations

Unconditionally stable regardless of damping parameter

Abstract

A third-order accurate implicit-explicit Runge-Kutta time marching numerical scheme is proposed and implemented for the Landau-Lifshitz-Gilbert equation, which models magnetization dynamics in ferromagnetic materials, with arbitrary damping parameters. This method has three remarkable advantages:~(1) only a linear system with constant coefficients needs to be solved at each Runge-Kutta stage, which greatly reduces the time cost and improves the efficiency; (2) the optimal rate convergence analysis does not impose any restriction on the magnitude of damping parameter, which is consistent with the third-order accuracy in time for 1-D and 3-D numerical examples; (3) its unconditional stability with respect to the damping parameter has been verified by a detailed numerical study. In comparison with many existing methods, the proposed method indicates a better performance on accuracy and efficiency, and thus provides a better option for micromagnetics simulations.