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

Yan Gui, Rui Du, Cheng Wang

TL;DR

This paper develops a third-order implicit-explicit Runge-Kutta (IMEX-RK3) time-stepping scheme for the Landau-Lifshitz-Gilbert equation with arbitrary damping. By introducing an artificial diffusion term and splitting the right-hand side into nonlinear (explicit) and linear (implicit) parts, the method requires only linear systems with constant coefficients at each RK stage, enabling fast Poisson-solvers. The authors establish an order-3 convergence theory that holds uniformly in the damping parameter, provide a stability analysis yielding unconditional stability with respect to $\alpha$, and demonstrate, through 1D and 3D numerical experiments, that the scheme achieves third-order temporal accuracy and second-order spatial accuracy while being robust to varying damping. The work offers a high-accuracy, efficient, and damping-parameter-insensitive tool for micromagnetics simulations and related spin dynamics problems.

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.

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

TL;DR

This paper develops a third-order implicit-explicit Runge-Kutta (IMEX-RK3) time-stepping scheme for the Landau-Lifshitz-Gilbert equation with arbitrary damping. By introducing an artificial diffusion term and splitting the right-hand side into nonlinear (explicit) and linear (implicit) parts, the method requires only linear systems with constant coefficients at each RK stage, enabling fast Poisson-solvers. The authors establish an order-3 convergence theory that holds uniformly in the damping parameter, provide a stability analysis yielding unconditional stability with respect to , and demonstrate, through 1D and 3D numerical experiments, that the scheme achieves third-order temporal accuracy and second-order spatial accuracy while being robust to varying damping. The work offers a high-accuracy, efficient, and damping-parameter-insensitive tool for micromagnetics simulations and related spin dynamics problems.

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.
Paper Structure (15 sections, 5 theorems, 107 equations, 2 figures, 7 tables)

This paper contains 15 sections, 5 theorems, 107 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The model and the proposed numerical method
  3. The Landau-Lifshitz equation
  4. Implicit-explicit Runge-Kutta methods
  5. Construction of IMEX RK3 scheme and its stability condition
  6. Order conditions
  7. Construction of IMEX RK3 scheme
  8. Stability condition
  9. Some preliminary inequalities
  10. Convergence analysis for the proposed IMEX-RK3 scheme
  11. Convergence analysis of the IMEX-RK3 scheme \ref{['IMEX-RK3']} for the nonlinear LL equation
  12. Numerical results
  13. Accuracy test of IMEX-RK3
  14. Dependence on the damping parameter
  15. Conclusions

Key Result

Lemma 4.1

For any grid functions $\boldsymbol f_h$ and $\boldsymbol g_h$, with $\boldsymbol f_h$ satisfying the discrete boundary condition eq-2, the following identity is valid:

Figures (2)

  • Figure 1: The 1-D spatial grids, where ${{x}_{-\frac{1}{2}}}$ and ${{x}_{N+\frac{1}{2}}}$ are two ghost points.
  • Figure 2: Temporal and spatial accuracy orders in the 1-D and 3-D domains computations. Top row: 1-D; Bottom row: 3-D.

Theorems & Definitions (9)

  • Remark 3.1
  • Remark 3.2
  • Definition 4.1: $\ell^2$ inner product, ${{\left\| \cdot \right\|}_{2}}$ norm
  • Definition 4.2: The discrete ${{\left\| \cdot \right\|}_{\infty}}$ and ${{\left\| \cdot \right\|}_{p}}$ norms
  • Lemma 4.1: Summation by parts
  • Lemma 4.2: Inverse inequality
  • Lemma 4.3: Discrete Gronwall inequality
  • Theorem 5.1
  • Lemma 5.1