  The numerical approximation for the Landau-Lifshitz equation, the dynamics of
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, with second-order accuracy
in both time and space, provided that the spatial step-size is the same order
as the temporal step-size. And also, the unique solvability of the numerical
solution is theoretically justified, which turns out to be the first such
result for the micromagnetics model. All these theoretical properties are
verified by numerical examples in both one- and three- dimensional spaces.

Analysis of PDEs;Numerical Analysis;Numerical Analysis