A Finite Difference method for the Wide-Angle `Parabolic' equation in a waveguide with downsloping bottom

TL;DR

This paper develops a stable and accurate finite difference method for solving the wide-angle parabolic equation in underwater acoustics with downsloping bottoms, addressing well-posedness issues and ensuring energy conservation.

Contribution

It introduces an additional boundary condition for downsloping bottoms to ensure well-posedness and develops a Crank-Nicolson finite difference scheme that is unconditionally stable and second-order accurate.

Findings

The method is unconditionally stable.

The scheme accurately simulates realistic underwater acoustics.

The boundary condition ensures well-posedness for downsloping profiles.

Abstract

We consider the third-order wide-angle `parabolic' equation of underwater acoustics in a cylindrically symmetric fluid medium over a bottom of range-dependent bathymetry. It is known that the initial-boundary-value problem for this equation may not be well posed in the case of (smooth) bottom profiles of arbitrary shape if it is just posed e.g. with a homogeneous Dirichlet bottom boundary condition. In this paper we concentrate on downsloping bottom profiles and propose an additional boundary condition that yields a well posed problem, in fact making it $L^2$-conservative in the case of appropriate real parameters. We solve the problem numerically by a Crank-Nicolson-type finite difference scheme, which is proved to be unconditionally stable and second-order accurate, and simulates accurately realistic underwater acoustic problems.