{
  "id": "2003.14399",
  "version": "v1",
  "published": "2020-03-28T03:18:58.000Z",
  "updated": "2020-03-28T03:18:58.000Z",
  "title": "Long time $\\mathcal H^s_α$ stability of a classical scheme for Cahn-Hilliard equation with polynomial nonlinearity",
  "authors": [
    "Wansheng Wang"
  ],
  "comment": "25 pages, 2 figures",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "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 $\\mathcal H^s_\\alpha$ ($s=1,2,3$) 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 $\\mathcal H^3_\\alpha$ bound is obtained on basis of the uniform $H^1$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-28T03:18:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65M12",
      "65P99",
      "35K55",
      "65Z05"
    ],
    "keywords": [
      "polynomial nonlinearity",
      "cahn-hilliard equation",
      "classical scheme",
      "long time behaviour",
      "fourier pseudospectral space approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}