Numerical approximations of Allen-Cahn and Cahn-Hilliard equations

Stability analyses and error estimates are carried out for a number of commonly usednumerical schemes for the Allen-Cahn and Cahn-Hilliard equations. It is shown thatall the schemes we considered are either unconditionally energy stable, orconditionally energy stable with reasonable stability conditions in thesemi-discretized versions. Error estimates for selected schemes with aspectral-Galerkin approximation are also derived. The stability analyses and errorestimates are based on a weak formulation thus the results can be easily extended toother spatial discretizations, such as Galerkin finite element methods, which arebased on a weak formulation.