  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.
