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

We 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 we transform the problem into one with horizontal bottom for which we establish an a priori H1 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 H1 estimates and optimal-order error bounds for a Crank-Nicolson scheme.