Finite difference schemes for the ''parabolic'' equation in a variable depth environment with a rigid bottom boundary condition (Q2719221)

The authors consider a linear Schrödinger-type partial differential equation, the ``parabolic'' equation of underwater acoustics, in a layer of water bounded below by a rigid bottom of variable topography. Using a change of depth variable technique, the authors first transform the problem into one with horizontal bottom, then they establish an a priori \(H^1\) estimate and prove an optimal-order error bound in the maximum norm for a Crank-Nicolson-type finite difference approximation of its solution. They also consider the same problem with an alternative rigid bottom boundary condition due to \textit{L. Abrahamsson} and \textit{H.-O. Kreiss} [Math. Methods Appl. Sci. 13, No. 5, 385-390 (1990; Zbl 0723.35009); Boundary conditions for the parabolic equation in a range-dependent duct, J. Acoust. Soc. Am. 87, 2438-2441 (1990)] and prove again a priori \(H^1\) estimates and optimal-order error bounds for a Crank-Nicolson scheme.