Solutions to the discrete Airy equation: application to parabolic equation calculations (Q1883481)

A variety of strategies to derive an approximation to the discrete transparent boundary condition (DTBC) for the Schrödinger equation with a linear potential term in the exterior domain are proposed. The derivation is based on the knowledge of the exact solution (including the asymptotics) to the discrete Airy equation. The approach has two advantages over the standard approach of discretizing the continuous TBC: higher accuracy and efficiency; while discretized TBCs have usually quadratic effort, the sum-of-exponential approximation to DTBCs has only linear effort. Moreover, a simple criteria to check the stability of the method is provided. This paper is organized as follows: first, a generalization of the discrete Airy equation is discussed \[ y_{m + 1} - 2y_m + y_{m - 1} - cmy_m = 0,\quad c = \left( {\Delta x} \right)^3 ,\quad x_m = m\Delta x,\quad y_m \simeq y\left( {x_m } \right) \] and it is shown how to find exact and asymptotic solutions. Afterwards the authors present an application to a problem arising in ''parabolic equation'' calculations in (underwater) acoustics and radar propagation in the troposphere: they construct a DTBC for a Schrödinger equation with a linear potential term, discuss different approaches and present an efficient implementation by the sum-of-exponentials ansatz. Afterwards the authors analyze the stability of the resulting numerical scheme. In this case the Laplace transformed Schrödinger equation can be viewed as a general Airy equation. Finally they illustrate the results with a numerical example from underwater acoustics.