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On an initial‐boundary value problem for a wide‐angle parabolic equation in a waveguide with a variable bottom
Author(s) -
Dougalis V. A.,
Sturm F.,
Zouraris G. E.
Publication year - 2009
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1097
Subject(s) - mathematics , boundary value problem , mathematical analysis , discretization , dirichlet boundary condition , boundary (topology) , mixed boundary condition , robin boundary condition , geometry
We consider the third‐order Claerbout‐type wide‐angle parabolic equation (PE) of underwater acoustics in a cylindrically symmetric medium consisting of water over a soft bottom B of range‐dependent topography. There is strong indication that the initial‐boundary value problem for this equation with just a homogeneous Dirichlet boundary condition posed on B may not be well‐posed, for example when B is downsloping. We impose, in addition to the above, another homogeneous, second‐order boundary condition, derived by assuming that the standard (narrow‐angle) PE holds on B, and establish a priori H 2 estimates for the solution of the resulting initial‐boundary value problem for any bottom topography. After a change of the depth variable that makes B horizontal, we discretize the transformed problem by a second‐order accurate finite difference scheme and show, in the case of upsloping and downsloping wedge‐type domains, that the new model gives stable and accurate results. We also present an alternative set of boundary conditions that make the problem exactly energy conserving; one of these conditions may be viewed as a generalization of the Abrahamsson–Kreiss boundary condition in the wide‐angle case. Copyright © 2008 John Wiley & Sons, Ltd.