  We study in this paper the accuracy and stability of partially and fully
implicit schemes for phase field modeling. Through theoretical and numerical
analysis of Allen-Cahn and Cahn-Hillard models, we investigate the potential
problems of using partially implicit schemes, demonstrate the importance of
using fully implicit schemes and discuss the limitation of energy stability
that are often used to evaluate the quality of a numerical scheme for
phase-field modeling. In particular, we make the following observations: 1. a
convex splitting scheme (CSS in short) can be equivalent to some fully implicit
scheme (FIS in short) with a much different time scaling and thus it may lack
numerical accuracy; 2. most implicit schemes (in discussions) are energy-stable
if the time-step size is sufficiently small; 3. a traditionally known
conditionally energy-stable scheme still possess an unconditionally
energy-stable physical solution; 4. an unconditionally energy-stable scheme is
not necessarily better than a conditionally energy-stable scheme when the time
step size is not small enough; 5. a first-order FIS for the Allen-Cahn model
can be devised so that the maximum principle will be valid on the discrete
level and hence the discrete phase variable satisfies $|u_h(x)|\le 1$ for all
$x$ and, furthermore, the linearized discretized system can be effectively
preconditioned by discrete Poisson operators.
