Computing the Dirichlet--Neumann Operator on a Cylinder

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 commonly used for 2D fluids, such as conformal mapping and boundary integral methods, fail to generalize suitably to three dimensions. 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 subproblems, a key ingredient of which is the use of Zernike polynomials for the circular cross-section instead of the traditional Bessel functions. This lends spectral accuracy to the method and allowing significant computational speed-up. We rigorously analyze the algorithm, prove its applicability to a wide class of problems, and demonstrate its effectiveness numerically.