Finite difference schemes for the "parabolic" equation in a variable depth environment with a rigid bottom boundary condition

We consider a linear, Schrodinger-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 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. We also consider the same problem with an alternative rigid bottom boundary condition due to Abrahamsson and Kreiss and prove again a priori H-1 estimates and optimal-order error bounds for a Crank Nicolson scheme.