Computation of 2D Stokes flows via lightning and AAA rational approximation

TL;DR

This paper introduces the LARS algorithm, combining lightning and AAA rational approximation, to efficiently compute accurate 2D Stokes flows in complex domains, extending previous methods to more general geometries.

Contribution

The paper presents a novel LARS algorithm that enables fast, high-accuracy computation of 2D Stokes flows in smooth and multiply-connected domains using rational approximation techniques.

Findings

Computations take less than a second with at least 6-digit accuracy.

The method is validated against analytical solutions.

It extends rational approximation techniques to general 2D Stokes flow problems.

Abstract

Low Reynolds number fluid flows are governed by the Stokes equations. In two dimensions, Stokes flows can be described by two analytic functions, known as Goursat functions. Brubeck and Trefethen (2022) recently introduced a lightning Stokes solver that uses rational functions to approximate the Goursat functions in polygonal domains. In this paper, we present the "LARS" algorithm (Lightning-AAA Rational Stokes) for computing 2D Stokes flows in domains with smooth boundaries and multiply-connected domains using lightning and AAA rational approximation (Nakatsukasa et al., 2018). After validating our solver against known analytical solutions, we solve a variety of 2D Stokes flow problems with physical and engineering applications. Using these examples, we show rational approximation can now be used to compute 2D Stokes flows in general domains. The computations take less than a second and give solutions with at least 6-digit accuracy.