Convergence analysis of a second-order semi-implicit projection method for Landau-Lifshitz equation

The numerical approximation for the Landau-Lifshitz equation, which models the dynamics of the magnetization in a ferromagnetic material, is taken into consideration. This highly nonlinear equation, with a non-convex constraint, has several equivalent forms, and involves solving an auxiliary problem in the infinite domain. All these features have posed interesting challenges in developing numerical methods. In this paper, we first present a fully discrete semi-implicit method for solving the Landau-Lifshitz equation based on the second-order backward differentiation formula and the one-sided extrapolation (using previous time-step numerical values). A projection step is further used to preserve the length of the magnetization. Subsequently, we provide a rigorous convergence analysis for the fully discrete numerical solution by the introduction of two sets of approximated solutions where one set of solutions solves the Landau-Lifshitz equation and the other is projected onto the unit sphere. Second-order accuracy in both time and space is obtained provided that the spatial step-size is the same order as the temporal step-size. And also, the unique solvability of the numerical solution without any assumption for the step-size in both time and space is theoretically justified, using a monotonicity analysis. All these theoretical properties are verified by numerical examples in both 1D and 3D spaces.