Finite difference methods for the wide-angle "parabolic" equation

We consider a model initial and boundary value problem for the wide-angle " parabolic" equation Lu-r = icu of underwater acoustics, where L is a second -order differential operator in the depth variable z with depth- and range- dependent coefficients. We discretize the problem by the Crank-Nicolson fin ite difference scheme and also by the forward Euler method using nonuniform partitions both in depth and in range. Assuming that the problem admits a smooth solution, and L is invertible for all r under the posed boundary and interface conditions, we show stability of both schemes and derive error e stimates.