(untitled) (Communications in Computational Physics, 2024)
In this paper, we investigate two fundamental geometric properties of the Landau-Lifshitz-Gilbert (LLG) equation, namely the preservation of magnetization magnitude and the Lyapunov structure, by using multistep methods. While the majority of current multistep methods for solving the LLG equation are based on two-step discrete schemes, our research specifically focuses on investigating more general multistep methods. Our proposed methods encompass a range of multistep discrete schemes that allow for achieving any desired order of accuracy in the temporal domain. In this highly general framework, we demonstrate that the magnitude of magnetization is preserved within an error of order (p+2) in time when employing a (p+1)th-order multistep discrete scheme. Additionally, the Lyapunov structure is preserved with a first-order error of temporal step size. Finally, some numerical experiments are presented to validate the accuracy of the proposed multistep discrete schemes.