Error Estimates for Finite Difference Methods for a Wide-Angle "Parabolic" Equation

We consider a model initial and boundary value problem for a third-order partial differential equation (PDE), a wide-angle "parabolic" equation frequently used in underwater acoustics, with depth- and range-dependent coefficients in the presence of horizontal interfaces and dissipation. After commenting on the existence-uniqueness theory of solution of the equation, we discretize the problem by a second-order finite difference method of Crank-Nicolson type for which we prove stability and optimal-order error estimates in suitable discrete L2- H1-, and maximum norms. We also prove, under certain conditions, that the forward Euler scheme is also stable and convergent for the problem at hand.