A finite difference method for the wide‐angle “parabolic” equation in a waveguide with downsloping bottom |
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| Authors: | Dimitra C Antonopoulou Vassilios A Dougalis Georgios E Zouraris |
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| Institution: | 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;3. Department of Mathematics, University of Athens, Panepistimiopolis, GR–157 84 Zographou, Greece;4. Department of Mathematics, University of Crete, GR–714 09 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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| Keywords: | downsloping bottom error estimates finite difference methods initial‐boundary‐value problems underwater sound propagation variable domains wide‐angle parabolic equation |
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