A localized absorbing boundary condition for discretized parabolic equation

Abstract

Terrains that ensure uni-directional propagation lead to reducing the governing Helmholtz wave equation to Parabolic Equation (PE) if the ray angles are restricted to be narrow (within ±15°) to the axis of propagation and the paraxial refractive index inhomogeneity is smooth [6]. Transparent Boundary Conditions (TBC) need to be placed to achieve domain truncation and there are two approaches for the derivation of the TBCs. In the first approach, a continuous TBC is first derived, which is then approximated for the discretization scheme [5], [2]. Reference [3] showed that this approach cannot assure unconditional stability when the finite difference (FD) discretization does not match the discretization of the continuous TBC. Alternatively, one could start directly from the Crank-Nicholson FD discretization, done by Ehrhardt and Arnold [3], and derive numerically exact discrete transparent boundary condition (DTBC). In either of these approaches, the TBC involves field values all the way to the initial plane, thus resulting in a non-local boundary condition (BC). In this paper we truncate this boundary condition, and employ a rational approximation to the spectral transfer function, thereby resulting in a localized absorbing boundary condition (ABC).