2024/10/25 by David P. Nicholls, Nicholls, David P., Jon Wilkening +3 · 1 citation
Computer Science · Mathematics · #35J40 #65N12 #65N35 #76B07 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #G.1.8 #Numerical Analysis (math.NA) #Spectral Theory in Mathematical Physics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2410.19296
openalex publication_date 2024/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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.