Landau–Lifshitz–Gilbert equation written in a suitable form for numerical solution via explicit solvers
Abstract
Several previous works have discussed specialised numerical methods for time integration of the Landau–Lifshitz–Gilbert equation. These methods were introduced with the principal aim of conserving the norm of the magnetisation vector, M . Recently, also, the addition of a norm-conserving term was proposed to facilitate the use of standard explicit numerical solvers. Here we derive a different form of the Landau–Lifshitz–Gilbert equation that can control numerical errors in ‖ M ‖ when integrated with explicit solvers, subject to the anti-alignment limitation discussed below. This form of the Landau–Lifshitz–Gilbert equation does not require the explicit addition of the norm-conserving term, since it naturally contains a similar term that numerically stabilises the solution to conserve ‖ M ‖ , provided M does not remain anti-parallel to the effective magnetic field for sustained times. Numerical tests show that, in ordinary circumstances, the error in ‖ M ‖ is controlled with slightly improved efficiency, compared to either adding the norm-conserving term explicitly or making use of step-wise renormalisation, as is typically done in micromagnetic codes. It is only for the extreme test case, of a single magnetisation vector which is initially anti-parallel to a static effective magnetic field, that the different form of the Landau–Lifshitz–Gilbert equation fails to conserve ‖ M ‖ , numerically.
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BibTeX
@article{botha2026landaulifshitzgilbert,
title = {{Landau–Lifshitz–Gilbert equation written in a suitable form for numerical solution via explicit solvers}},
author = {André E. Botha},
journal = {Scientific Reports},
year = {2026},
month = sep,
doi = {10.1038/s41598-026-73610-8},
url = {https://www.nature.com/articles/s41598-026-73610-8},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Landau–Lifshitz–Gilbert equation written in a suitable form for numerical solution via explicit solvers
AU - André E. Botha
PY - 2026
DA - 2026/09/30
JO - Scientific Reports
DO - 10.1038/s41598-026-73610-8
UR - https://www.nature.com/articles/s41598-026-73610-8
AB - Several previous works have discussed specialised numerical methods for time integration of the Landau–Lifshitz–Gilbert equation. These methods were introduced with the principal aim of conserving the norm of the magnetisation vector, M . Recently, also, the addition of a norm-conserving term was proposed to facilitate the use of standard explicit numerical solvers. Here we derive a different form of the Landau–Lifshitz–Gilbert equation that can control numerical errors in ‖ M ‖ when integrated with explicit solvers, subject to the anti-alignment limitation discussed below. This form of the Landau–Lifshitz–Gilbert equation does not require the explicit addition of the norm-conserving term, since it naturally contains a similar term that numerically stabilises the solution to conserve ‖ M ‖ , provided M does not remain anti-parallel to the effective magnetic field for sustained times. Numerical tests show that, in ordinary circumstances, the error in ‖ M ‖ is controlled with slightly improved efficiency, compared to either adding the norm-conserving term explicitly or making use of step-wise renormalisation, as is typically done in micromagnetic codes. It is only for the extreme test case, of a single magnetisation vector which is initially anti-parallel to a static effective magnetic field, that the different form of the Landau–Lifshitz–Gilbert equation fails to conserve ‖ M ‖ , numerically.
SN - 2045-2322
ER -