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

David P. Nicholls, Jon Wilkening, Xinyu Zhao

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.

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

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.

Paper Structure

This paper contains 24 sections, 8 theorems, 194 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The Water Wave Problem
  3. Quasiperiodic Functions
  4. The Dirichlet--Neumann Operator
  5. Transparent Boundary Conditions
  6. Analyticity of the Dirichlet--Neumann Operator
  7. Change of Variables
  8. Function Spaces
  9. Elliptic Estimate
  10. A Recursive Lemma
  11. Analyticity of the Field
  12. Analyticity of the DNO
  13. Numerical Algorithms
  14. High--Order Spectral Methods
  15. The Method of Operator Expansions
  16. ...and 9 more sections

Key Result

Lemma 4.2

Given an integer $s > d/2$ there exists a constant $M = M(s)$ such that all of the following estimates are true.

Figures (7)

  • Figure 1: Plot of relative error, \ref{['Eqn:RelErr']}, for a small perturbation ($\varepsilon=0.02$) in the surface Neumann data for the three HOPS algorithms (OE, FE, TFE) using both Taylor and Padé summation. Physical parameters were \ref{['Eqn:Params:Phys']} and numerical discretization was \ref{['Eqn:Params:Num']}.
  • Figure 2: Plot of relative error, \ref{['Eqn:RelErr']}, for a medium perturbation ($\varepsilon=0.5$) in the surface Neumann data for the three HOPS algorithms (OE, FE, TFE) using both Taylor and Padé summation. Physical parameters were \ref{['Eqn:Params:Phys']} and numerical discretization was \ref{['Eqn:Params:Num']}.
  • Figure 3: Plot of relative error, \ref{['Eqn:RelErr']}, for a large perturbation ($\varepsilon=1$) in the surface Neumann data for the three HOPS algorithms (OE, FE, TFE) using both Taylor and Padé summation. Physical parameters were \ref{['Eqn:Params:Phys']} and numerical discretization was \ref{['Eqn:Params:Num']}.
  • Figure 4: Plot of relative error, \ref{['Eqn:RelErr']}, in the surface Neumann data for the TFE algorithms using Taylor summation for perturbation sizes $\varepsilon = 0.01, 0.1, 0.2$. Physical parameters were \ref{['Eqn:Params:Phys:3d']} and numerical discretization was \ref{['Eqn:Params:Num:3d']}.
  • Figure 5: Plot of the computed Neumann data, $\tilde{u}$, for $\varepsilon=0.16$. (a) The full field; (b) A slice of the full field with a plane that has normal vector $(1/\sqrt{2}, -1/\sqrt{3}, -1)^T$.
  • ...and 2 more figures

Theorems & Definitions (23)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 4.1
  • Lemma 4.2
  • proof
  • Remark 4.3
  • Corollary 4.4
  • Lemma 4.5
  • Definition 4.6
  • ...and 13 more