  Magnetization dynamics in magnetic materials is often modeled by the
Landau-Lifshitz equation, which is solved numerically in general. In
micromagnetic simulations, the computational cost relies heavily on the
time-marching scheme and the evaluation of stray field. Explicit marching
schemes are efficient but suffer from severe stability constraints, while
nonlinear systems of equations have to be solved in implicit schemes though
they are unconditionally stable. A better compromise between stability and
efficiency is the semi-implicit scheme, such as the Gauss-Seidel projection
method (GSPM) and the second-order backward differentiation formula scheme
(BDF2). At each marching step, GSPM solves several linear systems of equations
with constant coefficients and updates the stray field several times, while
BDF2 updates the stray field only once but solves a larger linear system of
equations with variable coefficients and a nonsymmetric structure. In this
work, we propose a new method, dubbed as GSPM-BDF2, by combing the advantages
of both GSPM and BDF2. Like GSPM, this method is first-order accurate in time
and second-order accurate in space, and is unconditionally stable with respect
to the damping parameter. However, GSPM-BDF2 updates the stray field only once
per time step, leading to an efficiency improvement of about $60\%$ than the
state-of-the-art GSPM for micromagnetic simulations. For Standard Problem \#4
and \#5 from National Institute of Standards and Technology, GSPM-BDF2 reduces
the computational time over the popular software OOMMF by $82\%$ and $96\%$,
respectively. Thus, the proposed method provides a more efficient choice for
micromagnetic simulations.
