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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.