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

TL;DR

This paper introduces a new unconditionally convergent, second-order time integrator for the Landau-Lifshitz-Gilbert equation and its extension with eddy currents, requiring only linear solves per step.

Contribution

It presents a novel linear second-order IMEX integrator for LLG and ELLG equations, improving stability and efficiency over existing methods.

Findings

Unconditionally convergent scheme for LLG and ELLG.

Requires only one or two linear solves per time step.

Achieves almost second-order accuracy in time.

Abstract

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.