Characterizing the stabilization size for semi-implicit Fourier-spectral method to phase field equations
Abstract
Recent results in the literature provide computational evidence that stabilized semi-implicit time-stepping method can efficiently simulate phase field problems involving fourth-order nonlinear dif- fusion, with typical examples like the Cahn-Hilliard equation and the thin film type equation. The up-to-date theoretical explanation of the numerical stability relies on the assumption that the deriva- tive of the nonlinear potential function satisfies a Lipschitz type condition, which in a rigorous sense, implies the boundedness of the numerical solution. In this work we remove the Lipschitz assumption on the nonlinearity and prove unconditional energy stability for the stabilized semi-implicit time-stepping methods. It is shown that the size of stabilization term depends on the initial energy and the perturba- tion parameter but is independent of the time step. The corresponding error analysis is also established under minimal nonlinearity and regularity assumptions.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2014
- DOI:
- 10.48550/arXiv.1410.7541
- arXiv:
- arXiv:1410.7541
- Bibcode:
- 2014arXiv1410.7541L
- Keywords:
-
- Mathematics - Numerical Analysis
- E-Print:
- Siam J. Numer . Anal . Vol. 54, No. 3, pp. 1653-1681, 2016