Stable rational approximations for parabolic equation methods

TL;DR

This paper introduces the adaptive AAA rational approximation method for parabolic equations in wave propagation, demonstrating improved stability and efficiency over traditional methods in fluid-elastic waveguides.

Contribution

It applies the AAA algorithm to parabolic equations, showing enhanced stability and computational efficiency in waveguide simulations compared to existing approaches.

Findings

AAA approximations match reference solutions closely

AAA enables efficient split-step Padé method application

AAA outperforms rotated operator methods in stability

Abstract

Modern parabolic equation (PE) methods for wave propagation rely on application of a variety of fractional-powered differential operators. Rational approximations of these operators need to properly map their spectra onto the complex plane, accurately handling propagating modes while annihilating evanescent ones. Standard approaches for stable and accurate rational approximations include rotating the branch cut of the operators or imposing stability constraint equations, and have yielded accurate results for wave propagation in a variety of fluid, elastic, and fluid-elastic waveguides. The stability constraint method, however, does not yield operators that are stable for all fluid-elastic waveguides, and a recent study of waveguides comprised of a thin elastic layer overlaying a thick fluid layer revealed instabilities in the approximations derived from rotated operators. In this paper, we demonstrate the applicability of a different rational approximation method, the recently-developed adaptive Antoulas-Anderson (AAA) algorithm, to simulations of wave propagation using the fluid-elastic parabolic equation. We find that simulations using operators approximated using the AAA algorithm provide excellent agreement with reference solutions, with errors in transmission loss comparable to, and often less than, that of simulations using the rotated operator method. In addition, we find that the AAA algorithm allows for the application of the split-step Pad\'e method to fluid-elastic waveguides, which yields a large gain in computational efficiency.