• downsloping bottom;
  • error estimates;
  • finite difference methods;
  • initial-boundary-value problems;
  • underwater sound propagation;
  • variable domains;
  • wide-angle parabolic equation


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