Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. (English) Zbl 1201.65184

Shen, Jie; Yang, Xiaofeng

Discrete Contin. Dyn. Syst. 28, No. 4, 1669-1691 (2010).

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 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.

Reviewer: Wilhelm Heinrichs (Essen)

MSC:

65M70	Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs
35Q35	PDEs in connection with fluid mechanics
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M20	Method of lines for initial value and initial-boundary value problems involving PDEs
65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs

Keywords:

Allen-Cahn equation; Cahn-Hilliard equations; spectral method; error analysis; stability; semidiscretization; spectral-Galerkin approximation; Galerkin finite element methods