  Magnetization dynamics in ferromagnetic materials is modeled by the
Landau-Lifshitz (LL) equation, a nonlinear system of partial differential
equations. Among the numerical approaches, semi-implicit schemes are widely
used in the micromagnetics simulation, due to a nice compromise between
accuracy and efficiency. At each time step, only a linear system needs to be
solved and a projection is then applied to preserve the length of
magnetization. However, this linear system contains variable coefficients and a
non-symmetric structure, and thus an efficient linear solver is highly desired.
If the damping parameter becomes large, it has been realized that efficient
solvers are only available to a linear system with constant, symmetric, and
positive definite (SPD) structure. In this work, based on the implicit-explicit
Runge-Kutta (IMEX-RK) time discretization, we introduce an artificial damping
term, which is treated implicitly. The remaining terms are treated explicitly.
This strategy leads to a semi-implicit scheme with the following properties:
(1) only a few linear system with constant and SPD structure needs to be solved
at each time step; (2) it works for the LL equation with arbitrary damping
parameter; (3) high-order accuracy can be obtained with high-order IMEX-RK time
discretization. Numerically, second-order and third-order IMEX-RK methods are
designed in both the 1-D and 3-D domains. A comparison with the backward
differentiation formula scheme is undertaken, in terms of accuracy and
efficiency. The robustness of both numerical methods is tested on the first
benchmark problem from National Institute of Standards and Technology. The
linearized stability estimate and optimal rate convergence analysis are
provided for an alternate IMEX-RK2 numerical scheme as well.
