A finite difference method for the wide-angle “parabolic” equation in a waveguide with downsloping bottom

Authors

  • Dimitra C. Antonopoulou,

    1. Department of Applied Mathematics, University of Crete, GR–714 09 Heraklion, Greece
    2. Institute of Applied and Computational Mathematics, FO.R.T.H., GR–711 10 Heraklion, Greece
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  • Vassilios A. Dougalis,

    1. Institute of Applied and Computational Mathematics, FO.R.T.H., GR–711 10 Heraklion, Greece
    2. Department of Mathematics, University of Athens, Panepistimiopolis, GR–157 84 Zographou, Greece
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  • Georgios E. Zouraris

    Corresponding author
    1. Institute of Applied and Computational Mathematics, FO.R.T.H., GR–711 10 Heraklion, Greece
    2. Department of Mathematics, University of Crete, GR–714 09 Heraklion, Greece
    • Institute of Applied and Computational Mathematics, FO.R.T.H., GR–711 10 Heraklion, Greece
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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 article, we concentrate on downsloping bottom profiles and propose an additional boundary condition that yields a well-posed problem, in fact making it L2 -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. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

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