DISCRETE TRANSPARENT BOUNDARY CONDITIONS FOR WIDE ANGLE PARABOLIC EQUATIONS: FAST CALCULATION AND APPROXIMATION M. Ehrhardt and A. Arnold Matthias Ehrhardt, Technische Universit¨ at Berlin, Institut f¨ ur Mathematik, Str. des 17.Juni 136, D–10623 Berlin, Germany e-mail: ehrhardt@math.tu-berlin.de Anton Arnold, Institut f¨ ur Numerische Mathematik, Universit¨ at M¨ unster, Einsteinstr. 62, D–48149 M¨ unster, Germany e-mail: anton.arnold@math.uni-muenster.de This paper is concerned with the efficient implementation of transparent boundary condi- tions (TBCs) for wide angle parabolic equations (WAPEs) assuming cylindrical symmetry. In [1] a discrete TBC of convolution type was derived from the fully discretized whole–space problem that is reflection–free and yields an unconditionally stable scheme. Since the dis- crete TBC includes a convolution with respect to range with a weakly decaying kernel, its numerical evaluation becomes very costly for long-range simulations. As a remedy we construct new approximative transparent boundary conditions involving exponential sums as an approximation to the convolution kernel. This special approxima- tion enables us to use a fast evaluation of the convolution type boundary condition. This new approach was outlined in detail in [2] for the standard “parabolic” equation. 1. INTRODUCTION This paper is concerned with a finite difference discretization of wide angle “parabolic” equations. These models appear as one–way approximations to the Helmholtz equation in cylindrical coordinates with azimuthal symmetry. In particular we will discuss the efficient implementation of transparent boundary conditions. In oceanography one wants to calculate the underwater acoustic pressure p(z, r) emerg- ing from a time–harmonic point source located in the water at (z s , 0). Here, r > 0 denotes the radial range variable, 0 < z < z b is the depth variable. The water surface is at z = 0, and the (horizontal) sea bottom at z = z b . We denote the local sound speed by c(z, r), the density by ρ(z, r), and the attenuation by α(z, r) ≥ 0. n(z, r)= c 0 /c(z, r) is the refractive index, with a reference sound speed c 0 . Then the reference wave number is k 0 = 2π f /c 0 ,