2020/03/28 by Wansheng Wang, Wang, Wansheng
Computer Science · Materials Science · Mathematics · #35K55 #65M12 #65P99 #65Z05 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Navier-Stokes equation solutions #Numerical Analysis (math.NA) #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.2003.14399
openalex publication_date 2020/03/28 · openalex created_date 2020/04/10 · openalex updated_date 2026/07/28
In this paper we investigate the long time stability of the implicit Euler scheme for the Cahn-Hilliard equation with polynomial nonlinearity. The uniform estimates in H-1 and \mathcal Hsα (s=1,2,3) spaces independent of the initial data and time discrete step-sizes are derived for the numerical solution produced by this classical scheme with variable time step-sizes.The uniform \mathcal H3α bound is obtained on basis of the uniform H1 estimate for the discrete chemical potential which is derived with the aid of the uniform discrete Gronwall lemma. A comparison with the estimates for the continuous-in-time dynamical system reveals that the classical implicit Euler method can completely preserve the long time behaviour of the underlying system. Such a long time behaviour is also demonstrated by the numerical experiments with the help of Fourier pseudospectral space approximation.