Rational interpolation of the one-way Helmholtz propagator

This study is devoted to the higher-order finite-difference numerical methods for solving the pseudo-differential parabolic equation of diffraction theory. The relationship between the pseudo-differential propagation operator, variations of the refractive index, and the maximum propagation angle is established. It is proposed to use a rational approximation of the exponential propagation on an interval instead of the Pade approximation in a vicinity of a point. It is shown that using the approximation on an interval is more natural for this problem and allows using a more sparse computational grid than when using the local Pade approximation. Three variants of rational approximation on interval was considered: Chebyshev-Pade, rational interpolation and AAA method. The advantages and disadvantages of each approach are analyzed in the context of the problem under consideration. The proposed method differs from the existing ones only in the coefficients of the numerical scheme and does not require any significant changes in the implementations of the existing numerical schemas. Stability and error analysis of the proposed approach was carried out. The application of the proposed approach to the tropospheric radio-wave propagation and underwater acoustics is provided. Numerical examples and comparison with alternative methods quantitatively demonstrate the advantages of the proposed approach. Python 3 implementation of the proposed method is freely available. This work is an extended version of the ICCS-2021 conference paper (Lytaev, 2021).

Automated summary

This study focuses on higher-order finite-difference numerical methods for solving the pseudo-differential parabolic equation in diffraction theory. It proposes using a rational approximation of exponential propagation on an interval instead of Pade approximation near a point. The advantages and disadvantages of three variants of rational approximation on the interval are analyzed, and the proposed method is shown to have advantages in the context of tropospheric radio-wave propagation and underwater acoustics. The study provides numerical examples and comparisons to demonstrate the benefits of the proposed approach.