{
  "id": "1110.3438",
  "version": "v1",
  "published": "2011-10-15T20:19:27.000Z",
  "updated": "2011-10-15T20:19:27.000Z",
  "title": "A Finite Difference method for the Wide-Angle `Parabolic' equation in a waveguide with downsloping bottom",
  "authors": [
    "D. C. Antonopoulou",
    "V. A. Dougalis",
    "G. E. Zouraris"
  ],
  "comment": "2 figurew",
  "categories": [
    "math.NA"
  ],
  "abstract": "We consider the third-order wide-angle `parabolic' equation of underwater acoustics in a cylindrically symmetric fluid medium over a bottom of range-dependent bathymetry. It is known that the initial-boundary-value problem for this equation may not be well posed in the case of (smooth) bottom profiles of arbitrary shape if it is just posed e.g. with a homogeneous Dirichlet bottom boundary condition. In this paper we concentrate on downsloping bottom profiles and propose an additional boundary condition that yields a well posed problem, in fact making it $L^2$-conservative in the case of appropriate real parameters. We solve the problem numerically by a Crank-Nicolson-type finite difference scheme, which is proved to be unconditionally stable and second-order accurate, and simulates accurately realistic underwater acoustic problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-10-15T20:19:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65M06",
      "65M12",
      "65M15",
      "76Q05"
    ],
    "keywords": [
      "finite difference method",
      "wide-angle",
      "accurately realistic underwater acoustic problems",
      "simulates accurately realistic underwater acoustic",
      "crank-nicolson-type finite difference scheme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.3438A"
    }
  }
}