On Galerkin Methods for the Wide–angle Parabolic Equation

We consider the third–order, wide–angle, parabolic approximation of underwater acoustics in a medium with depth– and range–dependent speed of sound in the presence of dissipation and horizontal interfaces. We first discuss the theory of existence and uniqueness of solutions to the problem and derive an energy estimate. We then discretize the problem in the depth variable using two types of Galerkin/ finite element formulations that take into account the interface conditions, and in the range variable by the Crank–Nicolson and also a fourth–order accurate, implicit Runge–Kutta method. The resulting high–order numerical schemes are stable and convergent and are also shown to compare favorably with classical, implicit finite difference schemes in terms of computational effectiveness when applied to standard benchmark problems.