Stability analyses and error estimates are carried out for a number of commonly used
numerical schemes for the Allen-Cahn and Cahn-Hilliard equations. It is shown that
all the schemes we considered are either unconditionally energy stable, or
conditionally energy stable with reasonable stability conditions in the
semi-discretized versions. Error estimates for selected schemes with a
spectral-Galerkin approximation are also derived. The stability analyses and error
estimates are based on a weak formulation thus the results can be easily extended to
other spatial discretizations, such as Galerkin finite element methods, which are
based on a weak formulation.