{
  "id": "1707.09943",
  "version": "v1",
  "published": "2017-07-31T16:26:58.000Z",
  "updated": "2017-07-31T16:26:58.000Z",
  "title": "A \"converse\" stability condition is necessary for a compact higher order scheme on non-uniform meshes",
  "authors": [
    "Alexander Zlotnik",
    "Raimondas Čiegis"
  ],
  "comment": "6 pages, 2 figures, 1 table",
  "categories": [
    "math.NA"
  ],
  "abstract": "The stability bounds and error estimates for a compact higher order Numerov-Crank-Nicolson scheme on non-uniform space meshes for the 1D time-dependent Schr\\\"odinger equation have been recently derived. This analysis has been done in $L^2$ and $H^1$ mesh norms and used the non-standard \"converse\" condition $h_\\omega\\leq c_0\\tau$, where $h_\\omega$ is the mean space step, $\\tau$ is the time step and $c_0>0$. Now we prove that such condition is necessary for some families of non-uniform meshes and any space norm. Also numerical results show unacceptably wrong behavior of numerical solutions (their dramatic mass non-conservation) when this condition is violated.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-31T16:26:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65M06",
      "65M12"
    ],
    "keywords": [
      "compact higher order scheme",
      "non-uniform meshes",
      "stability condition",
      "compact higher order numerov-crank-nicolson scheme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}