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Solutions to the Discrete Airy Equation: Application to Parabolic Equation Calculations

Ehrhardt, Matthias; Mickens, Ronald E.

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

In the case of the equidistant discretization of the Airy differential equation (``discrete Airy equation'') the exact solution can be found explicitly. This fact is used to derive a discrete transparent boundary condition (TBC) for a Schrödinger-type equation with linear varying potential, which can be used in ``parabolic equation'' simulations in (underwater) acoustics and for radar propagation in the troposphere. We propose different strategies for the discrete TBC and show an efficient implementation. Finally a stability proof for the resulting scheme is given. A numerical example in the application to underwater acoustics shows the superiority of the new discrete TBC.