Towards Preserving Geometric Properties of Landau-Lifshitz-Gilbert Equation Using Multistep Methods (Communications in Computational Physics, 2024)
In this paper, we investigate two fundamental geometric properties of theLandau-Lifshitz-Gilbert (LLG) equation, namely the preservation of magnetizationmagnitude 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 temporaldomain. In this highly general framework, we demonstrate that the magnitude ofmagnetization is preserved within an error of order $(p+2)$ in time when employinga $(p+1)$th-order multistep discrete scheme. Additionally, the Lyapunov structure ispreserved with a first-order error of temporal step size. Finally, some numerical experiments are presented to validate the accuracy of the proposed multistep discreteschemes.