Geometrical integration of Landau–Lifshitz–Gilbert equation based on the mid-point rule

Landau–Lifshitz–Gilbert (LLG) equation is the fundamental equation to describe magnetization vector field dynamics in microscale and nanoscale magnetic systems. This equation is highly nonlinear in nature and, for this reason, it is generally solved by using numerical techniques. In this paper, the mid-point rule time-stepping technique is applied to the numerical time integration of LLG equation and the relevant properties of the numerical scheme are discussed. The mid-point rule is an unconditionally stable and second order accurate scheme which preserves the fundamental geometrical properties of LLG dynamics. First, it exactly preserves the LLG property of conserving the magnetization magnitude at each spatial location. Second, for constant in time applied fields, it preserves the LLG Lyapunov structure, namely the fact that the free energy is a decreasing function of time. In addition, in the case of zero damping, the mid-point rule preserves the conservation of the system free energy. The above preservation properties are unconditionally valid, i.e. they are fulfilled for any value of the time-step. Finally, the LLG hamiltonian structure in the case of zero damping is preserved up to the third order terms with respect to the time-step. The main difficulty related to this scheme is the necessity of solving a large system of globally coupled nonlinear equations. This problem has been circumvented by using special and reasonably fast quasi-Newton iterative technique. The proposed numerical scheme is then tested on the standard micromagnetic problem no. 4. In the numerical computations, the spatial discretization is obtained by finite difference technique and the magnetostatic field is computed through the Fast Fourier Transform method.