Finite difference scheme for the Landau-Lifshitz equation

In this paper we propose a finite difference scheme for the Landau-Lifshitz equation and the Heisenberg equation. These equations describe the evolution of spin fields in continuum ferromagnetism and have the following properties: (i) length preserving, (ii) energy conservation or dissipation property. We showthat our scheme inherits these characteristic properties. The proposed scheme is implicit nonlinear, so we check an unique solvability of our schemes. Furthermore, we establish the error estimate for this problem. Finally, we demonstrate numerical examples in order to show the effectiveness of our scheme.