Finite Difference Methods for the Wide-angle ‘parabolic’ Equation

We consider a model initial and boundary value problem for the wide-angle ‘parabolic’ equation Lur = icu of underwater acoustics, where L is a second-order differential operator in the depth variable z with depthand range-dependent coefficients. We discretize the problem by the Crank–Nicolson finite 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 estimates. Dedicated to the memory of Prof. Dr. Günther Hämmerlin