FINITE ELEMENT DISCRETIZATION OF THE PARABOLIC EQUATION IN AN UNDERWATER VARIABLE BOTTOM ENVIRONMENT

摘要

We use the standard 'parabolic' approximation to the 2D Helmholtz equation of underwater acoustics to model long-range propagation of sound in the sea in a domain with a rigid bottom of variable topography The rigid bottom is modeled by an exact Neumann boundary condition and a paraxial approximation thereof due to Abrahamsson and Kreiss The problem is solved by a change of variables that flattens the bottom and, subsequently, by a fully discrete finite element-Crank-Nicolson scheme After outlining the main stability and convergence results for the numerical scheme, we present results of numerical experiments with various bottom topographies These suggest that an apparent singularity develops in finite range in a case of interest where the existence theory of the p d e. fails.