BDF2-type integrator for Landau-Lifshitz-Gilbert equation in micromagnetics: IMEX time-stepping and full effective field
Abstract
We extend the BDF2-type finite-element integrator proposed in [M. Aldé, M. Feischl, D. Praetorius; BDF2-type integrator for Landau-Lifshitz-Gilbert equation in micromagnetics: unconditional weak convergence to weak solutions. (Math. Comp., 2026, doi:10.1090/mcom/4210)] to the Landau-Lifshitz-Gilbert (LLG) equation with lower-order effective-field contributions and current-induced non-conservative torques arising in micromagnetic applications. To incorporate these additional terms while retaining a linear solve at each time step, we employ an implicit-explicit (IMEX) time splitting. Our main theorem proves unconditional weak convergence of the fully discrete approximations towards weak solutions of the LLG equation. The analysis removes the mild but artificial CFL-type restrictions required at the first and final time steps in [Aldé M., Feischl M., Praetorius D.; Math. Comp., 2026]. Furthermore, in the presence of the Dzyaloshinskii-Moriya interaction (DMI), the convergence is likewise unconditional and does not require the CFL-type restriction imposed in [G. Hrkac, C.-M. Pfeiler, D. Praetorius, M. Ruggeri, A. Segatti, and B. Stiftner. Convergent tangent plane integrators for the simulation of chiral magnetic skyrmion dynamics. (Adv. Comput. Math., 2019. doi: 10.1007/s10444- 019-09667-z)]. Numerical experiments support the theoretical findings.
Related
- An unconditionally convergent BDF2-type finite element integrator for the inertial Landau-Lifshitz-Gilbert equation
- An explicit third-order low-regularity integrator for the cubic nonlinear Schrödinger equation
- Asymptotic stability of Bloch wall for dispersive Landau-Lifshitz equation
- Finite-energy travelling waves for the easy-plane Landau-Lifshitz equation
- The stochastic Landau-Lifshitz-Baryakhtar equation in critical spaces
BibTeX
@misc{alde2026bdf2type,
title = {{BDF2-type integrator for Landau-Lifshitz-Gilbert equation in micromagnetics: IMEX time-stepping and full effective field}},
author = {Michele Aldé and Dirk Praetorius and Agus L. Soenjaya},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10291},
eprint = {2610.10291},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10291},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - BDF2-type integrator for Landau-Lifshitz-Gilbert equation in micromagnetics: IMEX time-stepping and full effective field
AU - Michele Aldé
AU - Dirk Praetorius
AU - Agus L. Soenjaya
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10291
UR - https://arxiv.org/abs/2610.10291
AB - We extend the BDF2-type finite-element integrator proposed in [M. Aldé, M. Feischl, D. Praetorius; BDF2-type integrator for Landau-Lifshitz-Gilbert equation in micromagnetics: unconditional weak convergence to weak solutions. (Math. Comp., 2026, doi:10.1090/mcom/4210)] to the Landau-Lifshitz-Gilbert (LLG) equation with lower-order effective-field contributions and current-induced non-conservative torques arising in micromagnetic applications. To incorporate these additional terms while retaining a linear solve at each time step, we employ an implicit-explicit (IMEX) time splitting. Our main theorem proves unconditional weak convergence of the fully discrete approximations towards weak solutions of the LLG equation. The analysis removes the mild but artificial CFL-type restrictions required at the first and final time steps in [Aldé M., Feischl M., Praetorius D.; Math. Comp., 2026]. Furthermore, in the presence of the Dzyaloshinskii-Moriya interaction (DMI), the convergence is likewise unconditional and does not require the CFL-type restriction imposed in [G. Hrkac, C.-M. Pfeiler, D. Praetorius, M. Ruggeri, A. Segatti, and B. Stiftner. Convergent tangent plane integrators for the simulation of chiral magnetic skyrmion dynamics. (Adv. Comput. Math., 2019. doi: 10.1007/s10444- 019-09667-z)]. Numerical experiments support the theoretical findings.
ER -