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1 - High-Order Perturbation of Surfaces Short Course: Boundary Value Problems

Published online by Cambridge University Press:  05 February 2016

By
David P. Nicholls
Edited by
Thomas J. Bridges ,
Mark D. Groves  and
David P. Nicholls
David P. Nicholls
Affiliation:
University of Illinois at Chicago, Chicago
Thomas J. Bridges
Affiliation:
University of Surrey
Mark D. Groves
Affiliation:
Universität des Saarlandes, Saarbrücken, Germany
David P. Nicholls
Affiliation:
University of Illinois, Chicago
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Summary

Abstract

In this lecture we introduce two classical High-Order Perturbation of Surfaces (HOPS) computational schemes in the simplified context of elliptic boundary value problems inspired by models in water waves. For the problem of computing Dirichlet–Neumann Operators (DNOs) for Laplace's equation, we outline Bruno & Reitich's method of Field Expansions (FE) and then describe Milder and Craig & Sulem's method of Operator Expansions (OE). We further show how these algorithms can be extended to three dimensions and finite depth, and describe how Padé approximation can be used as a method of numerical analytic continuation to realize enhanced performance and applicability through a series of numerical experiments.

Introduction

Calculus in general, and Partial Differential Equations (PDEs) in particular have long been recognized as the most powerful and successful mathematical modeling tool for engineering and science, and the study of surface water waves is no exception. With the advent of the modern computer in the 1950s, the possibility of numerical simulation of PDEs at last became a practical reality. The last 50–60 years has seen an explosion in the development and implementation of algorithms for this purpose, which are rapid, robust, and highly accurate. Among the myriad choices are:

  1. Finite Difference methods (e.g., [1–4]),

  2. Finite Element methods (Continuous and Discontinuous) (e.g., [5–8]),

  3. High-Order Spectral (Element) methods (e.g., [9–14]),

  4. Boundary Integral/Element methods (e.g., [15, 16]).

The class of High-Order Perturbation of Surfaces (HOPS) methods we describe here are a High-Order Spectral method, which is particularly well suited for PDEs posed on piecewise homogeneous domains. Such “layered media” problems abound in the sciences, e.g., in

  1. • free-surface fluid mechanics (e.g., the water wave problem),

  2. • acoustic waves in piecewise constant density media,

  3. • electromagnetic waves interacting with grating structures,

  4. • elastic waves in sediment layers.

For such problems these HOPS methods can be

  1. • highly accurate (error decaying exponentially as the number of degrees of freedom increases),

  2. • rapid (an order of magnitude fewer unknowns as compared with volumetric formulations),

  3. • robust (delivering accurate results for rather rough/large interface shapes).

However, these HOPS schemes are not competitive for problems with inhomogeneous domains and/or “extreme” geometries.

Type
Chapter
Information
Lectures on the Theory of Water Waves , pp. 1 - 18
Publisher: Cambridge University Press
Print publication year: 2016

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References

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