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G. Di Fratta, C.-M. Pfeiler, D. Praetorius, M. Ruggeri, B. Stiftner:
"Linear second-order IMEX-type integrator for the (eddy current) Landau-Lifshitz-Gilbert equation";
in: "ASC Report 27/2017", Vienna University of Technology, Wien, 2017, ISBN: 978-3-902627-10-0, 1 - 40.

Combining ideas from [Alouges et al. (Numer. Math., 128, 2014)] and [Praetorius et al. (Comput. Math. Appl., 2017)], we propose a numerical algorithm for the integration of the nonlinear and time-dependent Landau-Lifshitz-Gilbert (LLG) equation which is unconditionally convergent, formally (almost) second-order in time, and requires only the solution of one linear system per time-step. Only the exchange contribution is integrated implicitly in time, while the lower-order contributions like the computationally expensive stray field are treated explicitly in time. Then, we extend the scheme to the coupled system of the Landau-Lifshitz-Gilbert equation with the eddy current approximation of Maxwell equations (ELLG). Unlike existing schemes for this system, the new integrator is unconditionally convergent, (almost) second-order in time, and requires only the solution of two linear systems per time-step.

micromagnetism, finite elements, linear second-order time integration, implicit-explicit time-marching scheme, unconditional convergence

http://www.asc.tuwien.ac.at/preprint/2017/asc27x2017.pdf