Numerical integration of Landau-Lifshitz-Gilbert equation based on the midpoint rule

he midpoint rule time discretization technique is applied to Landau-Lifshitz-Gilbert (LLG) equation. The technique is unconditionally stable and second-order accurate. It has the important property of preserving the conservation of magnetization amplitude of LLG dynamics. In addition, for typical forms of the micromagnetic free energy, the midpoint rule preserves the main energy balance properties of LLG dynamics. In fact, it preserves LLG Lyapunov structure and, in the case of zero damping, the system free energy. All the above preservation properties are fulfilled unconditionally, namely, regardless of the choice of the time step. The proposed technique is then tested on the standard micromagnetic problem No. 4. In the numerical computations, the magnetostatic field is computed by the fast Fourier transform method, and the nonlinear system of equations connected to the implicit time-stepping algorithm is solved by special and reasonably fast quasi-Newton technique.