Error analysis of a linear numerical scheme for the Landau-Lifshitz equation with large damping parameters

A second-order accurate, linear numerical method is analyzed for the Landau-Lifshitz equation with large damping parameters. This equation describes the dynamics of magnetization, with a non-convexity constraint of unit length of the magnetization. The numerical method is based on the second-order backward differentiation formula in time, combined with an implicit treatment for the linear diffusion term from the harmonic mapping part and explicit extrapolation for the nonlinear terms. Afterward, a projection step is applied to normalize the numerical solution at a point-wise level. This numerical scheme has shown extensive advantages in the practical computations for the physical model with large damping parameters, which comes from the fact that only a linear system with constant coefficients (independent of both time and the updated magnetization) needs to be solved at each time step, and has greatly improved the numerical efficiency. Meanwhile, a theoretical analysis for this linear numerical scheme has not been available. In this paper, we provide a rigorous error estimate of the numerical scheme, in the discrete l & INFIN;(0,T;l2)& AND;l2(0,T;Hh1)$$ {\ell} circumflex {\infty}\left(0,T;{\ell} circumflex 2\right)\cap {\ell} circumflex 2\left(0,T;{H}_h circumflex 1\right) $$ norm, under suitable regularity assumptions and reasonable ratio between the time step size and the spatial mesh size. In particular, the projection operation is nonlinear, and a stability estimate for the projection step turns out to be highly challenging. Such a stability estimate is derived in details, which will play an essential role in the convergence analysis for the numerical scheme, if the damping parameter is greater than 3.