TY - JOUR

T1 - Linear second-order IMEX-type integrator for the (eddy current) Landau–Lifshitz–Gilbert equation

AU - Di Fratta, Giovanni

AU - Pfeiler, Carl-Martin

AU - Praetorius, Dirk

AU - Ruggeri, Michele

AU - Stiftner, Bernhard

PY - 2020/10/31

Y1 - 2020/10/31

N2 - Combining ideas from Alouges et al. (2014, A convergent and precise finite element scheme for Landau–Lifschitz–Gilbert equation. Numer. Math., 128, 407–430) and Praetorius et al. (2018, Convergence of an implicit-explicit midpoint scheme for computational micromagnetics. Comput. Math. Appl., 75, 1719–1738) 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 the solution of only 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 LLG equation with the eddy current approximation of Maxwell equations. Unlike existing schemes for this system, the new integrator is unconditionally convergent, (almost) second-order in time, and requires the solution of only two linear systems per time step.

AB - Combining ideas from Alouges et al. (2014, A convergent and precise finite element scheme for Landau–Lifschitz–Gilbert equation. Numer. Math., 128, 407–430) and Praetorius et al. (2018, Convergence of an implicit-explicit midpoint scheme for computational micromagnetics. Comput. Math. Appl., 75, 1719–1738) 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 the solution of only 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 LLG equation with the eddy current approximation of Maxwell equations. Unlike existing schemes for this system, the new integrator is unconditionally convergent, (almost) second-order in time, and requires the solution of only two linear systems per time step.

KW - finite elements

KW - implicit-explicit time-marching scheme

KW - linear second-order time integration

KW - micromagnetism

KW - unconditional convergence

UR - http://www.scopus.com/inward/record.url?scp=85097262224&partnerID=8YFLogxK

UR - https://arxiv.org/abs/1711.10715

U2 - 10.1093/imanum/drz046

DO - 10.1093/imanum/drz046

M3 - Article

AN - SCOPUS:85097262224

SN - 0272-4979

VL - 40

SP - 2802

EP - 2838

JO - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

IS - 4

ER -