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

We consider a linear, Schrodinger-type partial differential equation, the p arabolic equation of underwater acoustics, in a layer of water bounded belo w by a rigid bottom of variable topography. Using a change of depth variabl e technique we transform the problem into one with horizontal bottom for wh ich 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 appro ximation of its solution. We also consider the same problem with an alterna tive rigid bottom boundary condition due to Abrahamsson and Kreiss and prov e again a priori H-1 estimates and optimal-order error bounds for a Crank N icolson scheme.