  Magnetization dynamics in magnetic materials is modeled by the
Landau-Lifshitz-Gilbert (LLG) equation. In the LLG equation, the length of
magnetization is conserved and the system energy is dissipative. Implicit and
semi-implicit schemes have been used in micromagnetics simulations due to their
unconditional numerical stability. In more details, implicit schemes preserve
the properties of the LLG equation, but solve a nonlinear system of equations
per time step. In contrast, semi-implicit schemes only solve a linear system of
equations, while additional operations are needed to preserve the length of
magnetization. It still remains unclear which one shall be used if both
implicit and semi-implicit schemes are available. In this work, using the
implicit Crank-Nicolson (ICN) scheme as a benchmark, we propose to make this
implicit scheme semi-implicit. It can be proved that both schemes are
second-order accurate in space and time. For the unique solvability of
nonlinear systems of equations in the ICN scheme, we require that the temporal
step size scales quadratically with the spatial mesh size. It is numerically
verified that the convergence of the nonlinear solver becomes slower for larger
temporal step size and multiple magnetization profiles are obtained for
different initial guesses. The linear systems of equations in the semi-implicit
CN (SICN) scheme are unconditionally uniquely solvable, and the condition that
the temporal step size scales linearly with the spatial mesh size is needed in
the convergence of the SICN scheme. In terms of numerical efficiency, the SICN
scheme achieves the same accuracy as the ICN scheme with less computational
time. Based on these results, we conclude that a semi-implicit scheme is
superior to its implicit analog both theoretically and numerically, and we
recommend the semi-implicit scheme in micromagnetics simulations if both
methods are available.
