{
  "id": "2202.09055",
  "version": "v1",
  "published": "2022-02-18T07:38:35.000Z",
  "updated": "2022-02-18T07:38:35.000Z",
  "title": "Convergence analysis of a finite difference method for stochastic Cahn--Hilliard equation",
  "authors": [
    "Jialin Hong",
    "Diancong Jin",
    "Derui Sheng"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "This paper presents the convergence analysis of the spatial finite difference method (FDM) for the stochastic Cahn--Hilliard equation with Lipschitz nonlinearity and multiplicative noise. Based on fine estimates of the discrete Green function, we prove that both the spatial semi-discrete numerical solution and its Malliavin derivative have strong convergence order $1$. Further, by showing the negative moment estimates of the exact solution, we obtain that the density of the spatial semi-discrete numerical solution converges in $L^1(\\mathbb R)$ to the exact one. Finally, we apply an exponential Euler method to discretize the spatial semi-discrete numerical solution in time and show that the temporal strong convergence order is nearly $\\frac38$, where a difficulty we overcome is to derive the optimal H\\\"older continuity of the spatial semi-discrete numerical solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-18T07:38:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite difference method",
      "spatial semi-discrete numerical solution",
      "stochastic cahn-hilliard equation",
      "convergence analysis",
      "semi-discrete numerical solution converges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}