A Crank–Nicolson scheme for the Landau–Lifshitz equation without damping

An accurate and efficient numerical approach, based on a finite difference method with Crank–Nicolson time stepping, is proposed for the Landau–Lifshitz equation without damping. The phenomenological Landau–Lifshitz equation describes the dynamics of ferromagnetism. The Crank–Nicolson method is very popular in the numerical schemes for parabolic equations since it is second-order accurate in time. Although widely used, the method does not always produce accurate results when it is applied to the Landau–Lifshitz equation. The objective of this article is to enumerate the problems and then to propose an accurate and robust numerical solution algorithm. A discrete scheme and a numerical solution algorithm for the Landau–Lifshitz equation are described. A nonlinear multigrid method is used for handling the nonlinearities of the resulting discrete system of equations at each time step. We show numerically that the proposed scheme has a second-order convergence in space and time.