{
  "id": "2006.16861",
  "version": "v1",
  "published": "2020-06-30T14:52:01.000Z",
  "updated": "2020-06-30T14:52:01.000Z",
  "title": "A time-domain preconditioner for the Helmholtz equation",
  "authors": [
    "Christiaan C. Stolk"
  ],
  "comment": "22 pages, 3 figures, 1 table",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "Time-harmonic solutions to the wave equation can be computed in the frequency or in the time domain. In the frequency domain, one solves a discretized Helmholtz equation, while in the time domain, the periodic solutions to a discretized wave equation are sought, e.g. by simulating for a long time with a time-harmonic forcing term. Disadvantages of the time-domain method are that the solutions are affected by temporal discretization errors and that the spatial discretization cannot be freely chosen, since it is inherited from the time-domain scheme. In this work we address these issues. Given an indefinite linear system satisfying certain properties, a matrix recurrence relation is constructed, such that in the limit the exact discrete solution is obtained. By iterating a large, finite number of times, an approximate solution is obtained, similarly as in a time-domain method for the Helmholtz equation. To improve the convergence, the process is used as a preconditioner for GMRES, and the time-harmonic forcing term is multiplied by a smooth window function. The construction is applied to a compact-stencil finite-difference discretization of the Helmholtz equation, for which previously no time-domain solver was available. Advantages of the resulting solver are the relative simplicity, small memory requirement and reasonable computation times.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-30T14:52:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N22",
      "65F10"
    ],
    "keywords": [
      "helmholtz equation",
      "time-domain preconditioner",
      "time-harmonic forcing term",
      "time domain",
      "time-domain method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}