Finite Element Discretization of the ‘parabolic’ Equation in an Underwater Variable Bottom Environment

The standard ‘parabolic’ approximation to the Helmholtz equation is used in order to model long-range propagation of sound in the sea in the presence of cylindrical symmetry in a domain with a rigid bottom of variable topography. The rigid bottom is modeled by a homogeneous Neumann condition and a paraxial approximation thereof proposed by Abrahamsson and Kreiss. The resulting initial-boundary-value problems are transformed, by a change of variables, to equivalent ones in a horizontal strip and, subsequently, are solved numerically by fully discrete finite element-Crank-Nicolson schemes. L and H error estimates of optimal order are proved for the numerical schemes in the case of upsloping bottoms for the Neumann boundary condition and for general bottom topographies in the case of Abrahamsson-Kreiss conditions. Results of numerical experiments are presented in domains with various bottom topographies. Some of these provide motivation for an alternative implementation of the Neumann boundary condition that yields improved numerical approximations.