Finite Difference Schemes for the "Parabolic" Equation in a Variable Depth Environment with a Rigid Bottom Boundary Condition

We consider a linear, Schrödinger type p.d.e., 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 we transform the problem into one with horizontal bottom, for which we establish an a priori H estimate and prove an optimal-order error bound in the maximum norm for a Crank-Nicolson type finite difference approximation of its solution. We also consider the same problem with an alternative rigid bottom boundary condition due to Abrahamsson and Kreiss, and prove again a priori H estimates and optimal order error bounds for a Crank-Nicolson scheme.