  It is a very common practice to use semi-implicit schemes in various
computations, which treat selected linear terms implicitly and the nonlinear
terms explicitly. For phase-field equations, the principal elliptic operator is
treated implicitly to reduce the associated stability constraints while the
nonlinear terms are still treated explicitly to avoid the expensive process of
solving nonlinear equations at each time step. However, very few recent
numerical analysis is relevant to semi-implicit schemes, while "stabilized"
schemes have become very popular. In this work, we will consider semi-implicit
schemes for the Allen-Cahn equation with {\em general potential} function. It
will be demonstrated that the maximum principle is valid and the energy
stability also holds for the numerical solutions. This paper extends the result
of Tang \& Yang (J. Comput. Math., 34(5):471--481, 2016) which studies the
semi-implicit scheme for the Allen-Cahn equation with {\em polynomial
potentials}.
