Numer. Math. Theor. Meth. Appl. 16, 182-203 (2023)
A Second-Order Semi-Implicit Method for the Inertial Landau-Lifshitz-Gilbert Equation
Panchi Li1, Lei Yang2, Jin Lan3, Rui Du1,4,* and Jingrun Chen1,4,5,6
1 School of Mathematical Sciences, Soochow University, Suzhou 215006, China
2 School of Computer Science and Engineering, Macau University of Science and Technology, Macao SAR, China
3 Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, 92 Weijin Road, Tianjin 300072, China
4 Mathematical Center for Interdisciplinary Research, Soochow University, Suzhou 215006, China
5 Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou, Jiangsu 215123, China
6 School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China
* durui@suda.edu.cn
Abstract
Electron spins in magnetic materials have preferred orientations collectively and generate the macroscopic magnetization. Its dynamics spans over a wide range of timescales from femtosecond to picosecond, and then to nanosecond. The Landau-Lifshitz-Gilbert (LLG) equation has been widely used in micromagnetics simulations over decades. Recent theoretical and experimental advances have shown that the inertia of magnetization emerges at sub-picosecond timescales and contributes significantly to the ultrafast magnetization dynamics, which cannot be captured intrinsically by the LLG equation. Therefore, as a generalization, the inertial LLG (iLLG) equation is proposed to model the ultrafast magnetization dynamics. Mathematically, the LLG equation is a nonlinear system of parabolic type with (possible) degeneracy. However, the iLLG equation is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy, and exhibits more complicated structures. It behaves as a hyperbolic system at sub-picosecond timescales, while behaves as a parabolic system at larger timescales spanning from picosecond to nanosecond. Such hybrid behaviors impose additional difficulties on designing efficient numerical methods for the iLLG equation. In this work, we propose a second-order semi-implicit scheme to solve the iLLG equation. The second-order temporal derivative of magnetization is approximated by the standard centered difference scheme, and the first-order temporal derivative is approximated by the midpoint scheme involving three time steps. The nonlinear terms are treated semi-implicitly using one-sided interpolation with second-order accuracy. At each time step, the unconditionally unique solvability of the unsymmetric linear system is proved with detailed discussions on the condition number. Numerically, the second-order accuracy of the proposed method in both time and space is verified. At sub-picosecond timescales, the inertial effect of ferromagnetics is observed in micromagnetics simulations, in consistency with the hyperbolic property of the iLLG model; at nanosecond timescales, the results of the iLLG model are in nice agreements with those of the LLG model, in consistency with the parabolic feature of the iLLG model.
