Analyticity and Stable Computation of Dirichlet-Neumann Operators for Laplace's Equation under Quasiperiodic Boundary Conditions in Two and Three Dimensions

TL;DR

This paper investigates the analyticity and stable computation of Dirichlet-Neumann Operators under quasiperiodic boundary conditions for Laplace's equation, enabling more accurate simulation of water waves in 2D and 3D.

Contribution

It rigorously defines DNOs for quasiperiodic boundaries, proves their analyticity, and introduces a novel high-order stable algorithm for their simulation.

Findings

Proved analyticity of DNOs under smooth boundary perturbations.

Developed a new stable, high-order numerical algorithm.

Validated the algorithm through extensive testing.

Abstract

Dirichlet-Neumann Operators (DNOs) are important to the formulation, analysis, and simulation of many crucial models found in engineering and the sciences. For instance, these operators permit moving-boundary problems, such as the classical water wave problem (free-surface ideal fluid flow under the influence of gravity and capillarity), to be restated in terms of interfacial quantities, which not only eliminates the boundary tracking problem, but also reduces the problem dimension. While these DNOs have been the object of much recent study regarding their numerical simulation and rigorous analysis, they have yet to be examined in the setting of laterally quasiperiodic boundary conditions. The purpose of this contribution is to begin this investigation with a particular eye towards the problem of more realistically simulating two and three dimensional surface water waves. Here we not only carefully define the DNO with respect to these boundary conditions for Laplace's equation, but we also show the rigorous analyticity of these operators with respect to sufficiently smooth boundary perturbations. These theoretical developments suggest a novel algorithm for the stable and high-order simulation of the DNO, which we implement and extensively test.