C.-M. Pfeiler, D. Praetorius,M. Ruggeri, B. Stiftner:

"Convergence of a second order implicit-explicit tangent plane scheme for the Landau-Lifshitz-Gilbert equation";

Poster: 11th International Symposium on Hysteresis Modeling and Micromagnetics, Barcelona; 2017-05-29 - 2017-05-31.

We consider the numerical approximation of the Landau-Lifshitz-Gilbert (LLG) equation, which describes the dynamics of the magnetization in a ferromagnetic material. The numerical integration of the LLG equation poses several challenges: strong nonlinearities, a nonconvex pointwise constraint, an intrinsic energy law, which combines conservative and dissipative effects, as well as the presence of nonlocal field contributions, which prescribe the coupling with other partial differential equations (PDEs). We extend the tangent plane scheme from [Alouges et al., 2014] and propose an algorithm in which all the lower-order contributions, e.g., the expensive-to-compute stray field, are treated explicitly in time by using an Adams-Bashforth approach. The numerical integrator is computationally attractive in the sense that, after the first time-step, only one linear system has to be solved per time-step. Despite this modification, the resulting scheme still fulfills a formal second-order convergence in time. Under appropriate assumptions, the convergence towards a weak solution of the problem is unconditional, i.e., the numerical analysis of the scheme does not require any CFL-type condition on the time-step size and the spatial mesh size. One particular focus is on the efficient treatment of coupled systems, for which we show that an approach based on the decoupling of the time integration of the LLG equation and the coupled PDE is very attractive in terms of computational cost and still leads to time-marching algorithms that are (unconditionally) convergent and second-order in time. Numerical experiments underpin our theoretical findings and demonstrate the applicability of the method for the simulation of practically relevant problem sizes.

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