Linear second-order IMEX-type integrator for the (eddy current) Landau-Lifshitz-Gilbert equation
This work provides a more efficient and stable numerical method for simulating magnetization dynamics in micromagnetics, particularly for coupled eddy current problems.
The authors propose a numerical integrator for the Landau-Lifshitz-Gilbert (LLG) equation that is unconditionally convergent, second-order accurate in time, and requires only one linear solve per time step. They extend it to the eddy current LLG system, achieving similar properties with two linear solves per step.
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.