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Register a new userA second-order semi-implicit method for the inertial Landau-Lifshitz-Gilbert equation
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Panchi Li
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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.
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We consider the numerical approximation of the inertial Landau-Lifshitz-Gilbert (iLLG) equation, which describes the dynamics of the magnetization in ferromagnetic materials at subpicosecond time scales. We propose and analyze two fully discrete numerical schemes based on two different approaches: The first method is based on a reformulation of the problem as a linear constrained variational formulation for the time derivative of the magnetization. The second method exploits a reformulation of the problem as a first order system in time for the magnetization and the angular momentum. Both schemes are implicit, based on first-order finite elements, and the constructed numerical approximations satisfy the inherent unit-length constraint of iLLG at the vertices of the underlying mesh. For both schemes, we establish a discrete energy law and prove convergence of the approximations towards a weak solution of the problem. Numerical experiments validate the theoretical results and show the applicability of the methods for the simulation of ultrafast magnetic processes.
Magnetization dynamics in magnetic materials is modeled by the Landau-Lifshitz-Gilbert (LLG) equation. In the LLG equation, the length of magnetization is conserved and the system energy is dissipative. Implicit and semi-implicit schemes have been used in micromagnetics simulations due to their unconditional numerical stability. In more details, implicit schemes preserve the properties of the LLG equation, but solve a nonlinear system of equations per time step. In contrast, semi-implicit schemes only solve a linear system of equations, while additional operations are needed to preserve the length of magnetization. It still remains unclear which one shall be used if both implicit and semi-implicit schemes are available. In this work, using the implicit Crank-Nicolson (ICN) scheme as a benchmark, we propose to make this implicit scheme semi-implicit. It can be proved that both schemes are second-order accurate in space and time. For the unique solvability of nonlinear systems of equations in the ICN scheme, we require that the temporal step size scales quadratically with the spatial mesh size. It is numerically verified that the convergence of the nonlinear solver becomes slower for larger temporal step size and multiple magnetization profiles are obtained for different initial guesses. The linear systems of equations in the semi-implicit CN (SICN) scheme are unconditionally uniquely solvable, and the condition that the temporal step size scales linearly with the spatial mesh size is needed in the convergence of the SICN scheme. In terms of numerical efficiency, the SICN scheme achieves the same accuracy as the ICN scheme with less computational time. Based on these results, we conclude that a semi-implicit scheme is superior to its implicit analog both theoretically and numerically, and we recommend the semi-implicit scheme in micromagnetics simulations if both methods are available.
In this paper, we present a finite difference heterogeneous multiscale method for the Landau-Lifshitz equation with a highly oscillatory diffusion coefficient. The approach combines a higher order discretization and artificial damping in the so-called micro problem to obtain an efficient implementation. The influence of different parameters on the resulting approximation error is discussed. Numerical examples for both periodic as well as more general coefficients are given to demonstrate the functionality of the approach.
In this paper, we consider several possible ways to set up Heterogeneous Multiscale Methods for the Landau-Lifshitz equation with a highly oscillatory diffusion coefficient, which can be seen as a means to modeling rapidly varying ferromagnetic materials. We then prove estimates for the errors introduced when approximating the relevant quantity in each of the models given a periodic problem, using averaging in time and space of the solution to a corresponding micro problem. In our setup, the Landau-Lifshitz equation with highly oscillatory coefficient is chosen as the micro problem for all models. We then show that the averaging errors only depend on $varepsilon$, the size of the microscopic oscillations, as well as the size of the averaging domain in time and space and the choice of averaging kernels.
We investigate in details the inertial dynamics of a uniform magnetization in the ferromagnetic resonance (FMR) context. Analytical predictions and numerical simulations of the complete equations within the Inertial Landau-Lifshitz-Gilbert (ILLG) model are presented. In addition to the usual precession resonance, the inertial model gives a second resonance peak associated to the nutation dynamics provided that the damping is not too large. The analytical resolution of the equations of motion yields both the precession and nutation angular frequencies. They are function of the inertial dynamics characteristic time $tau$, the dimensionless damping $alpha$ and the static magnetic field $H$. A scaling function with respect to $alphataugamma H$ is found for the nutation angular frequency, also valid for the precession angular frequency when $alphataugamma Hgg 1$. Beyond the direct measurement of the nutation resonance peak, we show that the inertial dynamics of the magnetization has measurable effects on both the width and the angular frequency of the precession resonance peak when varying the applied static field. These predictions could be used to experimentally identify the inertial dynamics of the magnetization proposed in the ILLG model.
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