A semi-discrete scheme for the stochastic Landau–Lifshitz equation

Abstract

We propose a new convergent time semi-discrete scheme for the stochastic Landau-Lifshitz-Gilbert equation. The scheme is only linearly implicit and does not require the resolution of a nonlinear problem at each time step. Using a martingale approach, we prove the convergence in law of the scheme up to a subsequence.

Splitting Integrators for the Stochastic Landau--Lifshitz Equation

“…A classical strategy for designing geometric integrators are splitting approaches, which were applied to the deterministic LL-equation in [5,18]. The articles [2,3,16] also incorporate a splitting step but with a different motivation. To the best of our knowledge, splitting approaches in the spirit of classical Hamiltonian splitting (as in [20,34]) have not been used for the stochastic LL-equation.…”

“…the authors in [22] formulate an equivalent matrix ODE on the Lie group G = (SO (3)) N such that (2.7) is naturally preserved and the accuracy of their method is independent from (2.7). The discretization error can be optimized by choosing an optimal infinitesimal update map which determines the onestep numerical update.…”

“…In the second version, the subsystems containing the exchange parts and the exterior parts of (2.1) can be solved explicitly, and only the local, noninteracting effects of anisotropy energy have to be treated by an implicit integrator. In particular, for the case E(m) = E [exch] (m) as, e.g., treated in [3,6], the method is fully explicit. For both versions, the computational cost mainly depends on the number of spins, and the number of interactions described by the exchange matrix J does not influence the size of the nonlinear systems.…”

Second-order semi-implicit projection methods for micromagnetics simulations

“…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.…”

Weak solutions of the Stochastic Landau–Lifshitz–Gilbert Equations with nonzero anisotrophy energy

“…There is also some research about the numerical schemes of equation (1.5), such as Baňas, Brzeźniak, and Prohl [5], Baňas, Brzeźniak, Neklyudov, and Prohl [6], Baňas, Brzeźniak, Neklyudov, and Prohl [7], Goldys, Le, and Tran [16] and Alouges, de Bouard and Hocquet [4]. The last paper differs from all previous papers as it deals with the LLGEs in the so called Gilbert form, see [15] and [3] for some related deterministic results, and with an infinite dimensional noise (correlated in space).…”