{
  "id": "0901.1190",
  "version": "v1",
  "published": "2009-01-09T09:13:14.000Z",
  "updated": "2009-01-09T09:13:14.000Z",
  "title": "Modified energy for split-step methods applied to the linear Schrödinger equation",
  "authors": [
    "Arnaud Debussche",
    "Erwan Faou"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "We consider the linear Schr\\\"odinger equation and its discretization by split-step methods where the part corresponding to the Laplace operator is approximated by the midpoint rule. We show that the numerical solution coincides with the exact solution of a modified partial differential equation at each time step. This shows the existence of a modified energy preserved by the numerical scheme. This energy is close to the exact energy if the numerical solution is smooth. As a consequence, we give uniform regularity estimates for the numerical solution over arbitrary long time",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-01-09T09:13:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65P10",
      "37M15"
    ],
    "keywords": [
      "linear schrödinger equation",
      "split-step methods",
      "modified energy",
      "numerical solution",
      "arbitrary long time"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}