Computing the Dirichlet-Neumann Operator on a Cylinder

TL;DR

This paper introduces a spectral method using Zernike polynomials to efficiently compute the Dirichlet-Neumann operator in cylindrical geometries, enabling accurate simulations of 3D fluid problems.

Contribution

The study develops a Transformed Field Expansion method with Zernike polynomials for spectral accuracy in 3D cylindrical geometries, overcoming limitations of traditional 2D techniques.

Findings

Spectral accuracy achieved with Zernike polynomials.

Fast solver for Poisson equations in cylindrical geometry.

Method applicable to a wide class of fluid problems.

Abstract

The computation of the Dirichlet-Neumann operator for the Laplace equation is the primary challenge for the numerical simulation of the ideal fluid equations. The techniques used commonly for 2D fluids, such as conformal mapping and boundary integral methods, fail to generalize suitably to 3D. In this study, we address this problem by developing a Transformed Field Expansion method for computing the Dirichlet-Neumann operator in a cylindrical geometry with a variable upper boundary. This technique reduces the problem to a sequence of Poisson equations on a flat geometry. We design a fast and accurate solver for these sub-problems, a key ingredient being the use of Zernike polynomials for the circular cross-section instead of the traditional Bessel functions. This lends spectral accuracy to the method as well as allowing significant computational speed-up. We rigorously analyze the algorithm and prove its applicability to a wide class of problems before demonstrating its effectiveness numerically.