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Combining ideas from [Alouges et al. (Numer. Math., 128, 2014)] and[Praetorius et al. (Comput. Math. Appl., 2017)], we propose a numericalalgorithm for the integration of the nonlinear and time-dependentLandau-Lifshitz-Gilbert (LLG) equation which is unconditionally convergent,formally (almost) second-order in time, and requires only the solution of onelinear system per time-step. Only the exchange contribution is integratedimplicitly in time, while the lower-order contributions like thecomputationally expensive stray field are treated explicitly in time. Then, weextend the scheme to the coupled system of the Landau-Lifshitz-Gilbert equationwith the eddy current approximation of Maxwell equations (ELLG). Unlikeexisting schemes for this system, the new integrator is unconditionallyconvergent, (almost) second-order in time, and requires only the solution oftwo linear systems per time-step.

Linear second-order IMEX-type integrator for the (eddy current) Landau-Lifshitz-Gilbert equation