Finite difference scheme for the Landau–Lifshitz equation

Fuwa, Atsushi; Ishiwata, Tetsuya; Tsutsumi, Masayoshi

doi:10.1007/s13160-011-0054-9

Cited by 15 publications

(21 citation statements)

References 9 publications

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“…Remark 3. Note that the unique solvability of Scheme A and Scheme B does not impose any condition on k and h. This is in contrast with earlier results for the fully implicit schemes where k = O(h 2 ) is needed for the unique solution of the nonlinear system of equations at each time step; see [20] for example.…”

Section: Unconditionally Unique Solvabilitymentioning

confidence: 81%

“…There has been a continuous progress of developing numerical algorithms in the past few decades; see for example [6,7] and references therein. The spatial derivative is typically approximated by the finite element method (FEM) [8,9,10,11,12,13,14,15,16,17,18] and the finite difference method [19,20,21,22,23]. As for the temporal discretization, explicit schemes [15,24], fully implicit schemes [25,26,20], and semi-implicit schemes [19,27,28,29,30,31,32] have been extensively studied.…”

Section: Introductionmentioning

confidence: 99%

“…A nonlinear multigrid method is used to handle the nonlinearity at each time step in [33], and the fixed point iteration technique is used to deal with the nonlinearity in [34]. In [20], the existence and uniqueness of a solution to the nonlinear system is proved under the condition that the temporal stepsize be proportional to the square of the spatial gridsize. This, however, is contrary to the unconditional stability of implicit schemes.…”

Section: Introductionmentioning

confidence: 99%

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Second-order semi-implicit projection methods for micromagnetics simulations

Xie

Garcı́a-Cervera

Wang

et al. 2020

Journal of Computational Physics

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Micromagnetics simulations require accurate approximation of the magnetization dynamics described by the Landau-Lifshitz-Gilbert equation, which is nonlinear, nonlocal, and has a non-convex constraint, posing interesting challenges in developing numerical methods. In this paper, we propose two second-order semi-implicit projection methods based on the second-order backward differentiation formula and the second-order interpolation formula using the information at previous two temporal steps. Unconditional unique solvability of both methods is proved, with their second-order accuracy verified through numerical examples in both 1D and 3D. The efficiency of both methods is compared to that of another two popular methods. In addition, we test the robustness of both methods for the first benchmark problem with a ferromagnetic thin film material from National Institute of Standards and Technology.

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Section: Unconditionally Unique Solvabilitymentioning

confidence: 81%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Second-order semi-implicit projection methods for micromagnetics simulations

Xie

Garcı́a-Cervera

Wang

et al. 2020

Journal of Computational Physics

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“…Here the numerical stability of the original GSPM [15] was founded to be independent of gridsizes but depend on the damping parameter α. This issue was solved in [18] by replacing (14) and (16) with…”

Section: Gauss-seidel Projection Methods For Landau-lifshitz-gilbert Ementioning

confidence: 99%

“…Another issue is that the length of magnetization cannot be preserved and thus a projection step is needed. Implicit schemes [12,13,14] are unconditionally stable and usually can preserve the length of magnetization automatically. The difficulty of implicit schemes is how to solve a nonlinear system of equations at each step.…”

Section: Introductionmentioning

confidence: 99%

Two improved Gauss-Seidel projection methods for Landau-Lifshitz-Gilbert equation

Xie

et al. 2020

Journal of Computational Physics

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A B S T R A C TMicromagnetic simulation is an important tool to study various dynamic behaviors of magnetic order in ferromagnetic materials. The underlying model is the Landau-Lifshitz-Gilbert equation, where the magnetization dynamics is driven by the gyromagnetic torque term and the Gilbert damping term. Numerically, considerable progress has been made in the past decades. One of the most popular methods is the Gauss-Seidel projection method developed by Xiao-Ping Wang, Carlos García-Cervera, and Weinan E in 2001. It first solves a set of heat equations with constant coefficients and updates the gyromagnetic term in the Gauss-Seidel manner, and then solves another set of heat equations with constant coefficients for the damping term. Afterwards, a projection step is applied to preserve the length constraint in the pointwise sense. This method has been verified to be unconditionally stable numerically and successfully applied to study magnetization dynamics under various controls.In this paper, we present two improved Gauss-Seidel projection methods with unconditional stability. The first method updates the gyromagnetic term and the damping term simultaneously and follows by a projection step. The second method introduces two sets of approximate solutions, where we update the gyromagnetic term and the damping term simultaneously for one set of approximate solutions and apply the projection step to the other set of approximate solutions in an alternating manner. Compared to the original Gauss-Seidel projection method which has to solve heat equations 7 times at each time step, the improved methods solve heat equations 5 times and 3 times, respectively. First-order accuracy in time and second-order accuracy in space are verified by examples in both 1D and 3D. In addition, unconditional stability with respect to both the grid size and the damping parameter is confirmed numerically. Application of both methods to a realistic material is also presented with hysteresis loops and magnetization profiles. Compared with the original method, the recorded running times suggest that savings of both methods are about 2/7 and 4/7 for the same accuracy requirement, respectively.

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An Adaptive Step Implicit Midpoint Rule for the Time Integration of Newton’s Linearisations of Non-Linear Problems with Applications in Micromagnetics

et al. 2019

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The implicit mid-point rule is a Runge-Kutta numerical integrator for the solution of initial value problems, which possesses important properties that are relevant in micromagnetic simulations based on the Landau-Lifshitz-Gilbert equation, because it conserves the magnetization length and accurately reproduces the energy balance (i.e. preserves the geometric properties of the solution). We present an adaptive step size version of the integrator by introducing a suitable local truncation error estimator in the context of a predictor-corrector scheme. We demonstrate on a number of relevant examples that the selected step sizes in the adaptive algorithm are comparable to the widely used adaptive second-order integrators, such as the backward differentiation formula (BDF2) and the trapezoidal rule. The proposed algorithm is suitable for a wider class of non-linear problems, which are linearised by Newton's method and require the preservation of geometric properties in the numerical solution. Keywords Initial value problems • Runge-Kutta methods • Adaptive time integration • Predictor-corrector methods • Micromagnetics • Landau-Lifshitz-Gilbert equation 1 Background and Context Initial value problems (IVP) arise in mathematical models of many important physical and engineering processes and phenomena. They either appear as stand-alone problems (as ordinary differential equations (ODEs), where the unknown functions depend only on a single variable, for example describing the motion in classical mechanics or chemical reactions), or in the context of applying the method of lines to partial differential equations (PDE), i.e. the problems where unknown functions have both spatial and time variation [1, p. 8]. The solutions of such problems frequently exhibit multiple spatio-temporal scales. In such cases implicit solvers allow the deployment of larger step sizes, while maintaining stability. Efficient integrators should also involve adaptivity [2, p. 73,74], which can be realised both

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Finite difference scheme for the Landau–Lifshitz equation

Cited by 15 publications

References 9 publications

Second-order semi-implicit projection methods for micromagnetics simulations

Second-order semi-implicit projection methods for micromagnetics simulations

Two improved Gauss-Seidel projection methods for Landau-Lifshitz-Gilbert equation

An Adaptive Step Implicit Midpoint Rule for the Time Integration of Newton’s Linearisations of Non-Linear Problems with Applications in Micromagnetics

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