Convergence Analysis of A Secondorder Accurate, Linear Numerical Scheme for The LandauLifshitz Equation with Large Damping Parameters
Abstract
A second order accurate, linear numerical method is analyzed for the LandauLifshitz equation with large damping parameters. This equation describes the dynamics of magnetization, with a nonconvexity constraint of unit length of the magnetization. The numerical method is based on the secondorder backward differentiation formula in time, combined with an implicit treatment of the linear diffusion term and explicit extrapolation for the nonlinear terms. Afterward, a projection step is applied to normalize the numerical solution at a pointwise 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 $\ell^{\infty}(0,T; \ell^2) \cap \ell^2(0,T; H_h^1)$ norm, under suitable regularity assumptions and reasonable ratio between the time stepsize and the spatial meshsize. 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 arXiv:
 arXiv:2111.07537
 Bibcode:
 2021arXiv211107537C
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematical Physics;
 35K61;
 65M06;
 65M12