Based on the finite element method, a fully discrete Crank–Nicolson/leapfrog scheme is proposed and studied for the Landau–Lifshitz equation. The scheme avoids stiff time step restrictions and requires that the time step only be bounded by a constant independent of the mesh size. Then, almost unconditional optimal error estimates for the discrete scheme are obtained by using a temporal-spatial error splitting technique. The theoretical results are verified by numerical experiments.
"A LINEARIZED CRANK–NICOLSON/LEAPFROG SCHEME FOR THE LANDAU–LIFSHITZ EQUATION." Rocky Mountain J. Math. 53 (3) 821 - 837, June 2023. https://doi.org/10.1216/rmj.2023.53.821