  We propose a numerical integrator for the coupled system of the eddy-current
equation with the nonlinear Landau-Lifshitz-Gilbert equation. The considered
effective field contains a general field contribution, and we particularly
cover exchange, anisotropy, applied field, and magnetic field (stemming from
the eddy-current equation). Even though the considered problem is nonlinear,
our scheme requires only the solution of two linear systems per time-step.
Moreover, our algorithm decouples both equations so that in each time-step, one
linear system is solved for the magnetization, and afterwards one linear system
is solved for the magnetic field. Unconditional convergence -- at least of a
subsequence -- towards a weak solution is proved, and our analysis even
provides existence of such weak solutions. Numerical experiments with a
micromagnetic benchmark problem underline the performance of the proposed
algorithm.
