Spectral Methods for capillary surfaces described by bounded generating curves

TL;DR

This paper introduces a stable, efficient pseudo-spectral method using Chebyshev polynomials for solving boundary value problems related to capillary surfaces generated by bounded curves, including drops and tubes.

Contribution

It develops a novel pseudo-spectral approach for capillary surface equations that improves stability and computational efficiency over traditional methods.

Findings

The method is more stable than shooting methods.

It is computationally fast and lean.

The algorithm is adaptive without using Chebfun's automation.

Abstract

We consider capillary surfaces that are constructed by bounded generating curves. This class of surfaces includes radially symmetric and lower dimensional fluid-fluid interfaces. We use the arc-length representation of the differential equations for these surfaces to allow for vertical points and inflection points along the generating curve. These considerations admit capillary tubes, sessile drops, and fluids in annular tubes as well as other examples. We present a pseudo-spectral method for approximating solutions to the associated boundary value problems based on interpolation by Chebyshev polynomials. This method is observably more stable than the traditional shooting method and it is computationally lean and fast. The algorithm is also adaptive, but does not use the adaptive automation in Chebfun.