  Phase-field model is a powerful mathematical tool to study the dynamics of
interface and morphology changes in fluid mechanics and material sciences.
However, numerically solving a phase field model for a real problem is a
challenge task due to the non-convexity of the bulk energy and the small
interface thickness parameter in the equation. In this paper, we propose two
stabilized second order semi-implicit linear schemes for the Allen-Cahn
phase-field equation based on backward differentiation formula and
Crank-Nicolson method, respectively. In both schemes, the nonlinear bulk force
is treated explicitly with two second-order stabilization terms, which make the
schemes unconditional energy stable and numerically efficient. By using a known
result of the spectrum estimate of the linearized Allen-Cahn operator and some
regularity estimate of the exact solution, we obtain an optimal second order
convergence in time with a prefactor depending on the inverse of the
characteristic interface thickness only in some lower polynomial order. Both
2-dimensional and 3-dimensional numerical results are presented to verify the
accuracy and efficiency of proposed schemes.
