FINITE ELEMENT METHODS FOR THE PARABOLIC EQUATION WITH INTERFACES

Abstract

We consider the standard parabolic approximation of underwater acoustics in a medium consisting of fluid layers with depth- and range-dependent speed of sound and dissipation coefficients. In the case of horizontal layers we construct two types of finite element discretizations in the depth variable which treat the interfaces in different ways. We couple these methods with conservative Crank–Nicolson and implicit Runge–Kutta schemes in the range variable and analyze the stability and convergence of the resulting fully discrete methods. Using a change of variable technique, we show how these schemes may be extended to treat interfaces with range-dependent topography; we also couple them with a nonlocal ('impedance') bottom boundary condition. The methods are applied to several benchmark problems in the literature and their results are compared to those obtained from standard numerical codes.