A second-order semi-implicit method for the inertial Landau-Lifshitz-Gilbert equation

by   Panchi Li, et al.

Recent theoretical and experimental advances show that the inertia of magnetization emerges at sub-picoseconds and contributes to the ultrafast magnetization dynamics which cannot be captured intrinsically by the LLG equation. Therefore, as a generalization, the inertial Landau-Lifshitz-Gilbert (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 like a hyperbolic system at the sub-picosecond scale while behaves like a parabolic system at larger timescales. Such hybrid behaviors impose additional difficulties on designing numerical methods for the iLLG equation. In this work, we propose a second-order semi-implicit scheme to solve the iLLG equation. The second temporal derivative of magnetization is approximated by the standard centered difference scheme and the first derivative is approximated by the midpoint scheme involving three time steps. The nonlinear terms are treated semi-implicitly using one-sided interpolation with the second-order accuracy. At each step, the unconditionally unique solvability of the unsymmetric linear system of equations in the proposed method is proved with a detailed discussion on the condition number. Numerically, the second-order accuracy in both time and space is verified. Using the proposed method, the inertial effect of ferromagnetics is observed in micromagnetics simulations at small timescales, in consistency with the hyperbolic property of the model at sub-picoseconds. For long time simulations, 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 at larger timescales.


Convergence analysis of an implicit finite difference method for the inertial Landau-Lifshitz-Gilbert equation

The Landau-Lifshitz-Gilbert (LLG) equation is a widely used model for fa...

Second-order semi-implicit projection methods for micromagnetics simulations

Micromagnetics simulations require accurate approximation of the magneti...

Second-order flows for computing the ground states of rotating Bose-Einstein condensates

Second-order flows in this paper refer to some artificial evolutionary d...

Analysis of a WSGD scheme for backward fractional Feynman-Kac equation with nonsmooth data

In this paper, we propose and analyze a second-order time-stepping numer...

A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel

In this paper, a two-grid temporal second-order scheme for the two-dimen...

Proper Selection of Obreshkov-Like Numerical Integrators Used as Numerical Differentiators

Criteria for Obreshkov-like numerical integrators to be used as numerica...

Anomalous sorption kinetics of self-interacting particles by a spherical trap

In this paper we propose a computational framework for the investigation...

Please sign up or login with your details

Forgot password? Click here to reset