Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2025-01-03T16:02:19.596Z Has data issue: false hasContentIssue false

3 - High-Order Perturbation of Surfaces Short Course: Analyticity Theory

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
Get access

Summary

Abstract

In this contribution we take up the question of convergence of the classical High-Order Perturbation of Surfaces (HOPS) schemes we introduced in the first lecture. This is intimately tied to analyticity properties of the relevant fields and Dirichlet–Neumann Operators. We show how a straightforward approach cannot succeed. However, with a simple change of variables this very method delivers not only a clear and optimal analyticity theory, but also a stable and high-order numerical scheme. We justify this latter claim with representative numerical simulations involving all three of the HOPS schemes presented thus far.

Introduction

Over the past two lectures we have derived several High-Order Perturbation of Surfaces (HOPS) schemes for the numerical simulation of (i.) solutions to boundary value problems, (ii.) surface integral operators (e.g., the Dirichlet–Neumann Operator [DNO]), and (iii.) free and moving boundary problems (e.g., traveling water waves). These HOPS methods are rapid (amounting to surface formulations accelerated by FFTs), robust (for perturbation size sufficiently small), and simple to implement (Operator Expansions is a one-line formula, while Field Expansions (FE) is two lines!). The derivation of all of these schemes is based upon the analyticity of the unknowns, but we have not yet specified under what conditions this is true. In this contribution we describe a straightforward framework for addressing this question that can be extended to give the most generous hypotheses known.

The first result on analyticity properties of DNOs with respect to boundary perturbations is due to Coifman & Meyer [1]. In this work the DNO was shown to be analytic as a function of Lipschitz perturbations of a line in the plane. Using a different formulation Craig, Schanz, and Sulem [2] and Craig & Nicholls [3, 4] proved analyticity of the DNO for C1 perturbations of a hyperplane in three and general d dimensions, respectively. All of these results depend upon delicate estimates of integral operators appearing in surface formulations of the problem defining the DNO.

Using a completely different approach (which we describe here), the author and F. Reitich produced a greatly simplified approach to establishing analyticity of DNOs and their related fields (at the cost of slightly less generous hypotheses, C2 smoothness of the boundary perturbation) [5–7], which has the added benefit of generating a stabilized numerical approach.

Type
Chapter
Information
Lectures on the Theory of Water Waves , pp. 32 - 50
Publisher: Cambridge University Press
Print publication year: 2016

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Coifman, R., and Meyer, Y. 1985. Nonlinear Harmonic Analysis and Analytic Dependence. Pages 71-78 of: Pseudodifferential operators and applications (Notre Dame, Ind., 1984). Amer. Math. Soc., pp. 71-78.Google Scholar
[2] Craig, Walter, Schanz, Ulrich, and Sulem, Catherine. 1997. The Modulation Regime of Three-Dimensional Water Waves and the Davey-Stewartson System. Ann. Inst. Henri Poincare, 14, 615-667.Google Scholar
[3] Nicholls, David P. 1998. Traveling Gravity Water Waves in Two and Three Dimensions. Ph.D. thesis, Brown University.
[4] Craig, Walter, and Nicholls, David P. 2000. Traveling Two and Three Dimensional Capillary Gravity Water Waves. SIAM J. Math. Anal., 32(2), 323-359.CrossRefGoogle Scholar
[5] Nicholls, David P., and Reitich, Fernando. 2001a. A new approach to analyticity of Dirichlet-Neumann operators. Proc. Roy. Soc. Edinburgh Sect. A, 131(6), 1411-1433.CrossRefGoogle Scholar
[6] Nicholls, David P., and Reitich, Fernando. 2001b. Stability of High-Order Perturbative Methods for the Computation of Dirichlet-Neumann Operators. J. Comput. Phys., 170(1), 276-298.CrossRefGoogle Scholar
[7] Nicholls, David P., and Reitich, Fernando. 2003. Analytic Continuation of Dirichlet-Neumann Operators. Numer. Math., 94(1), 107-146.CrossRefGoogle Scholar
[8] Nicholls, David P., and Reitich, Fernando. 2004a. Shape Deformations in Rough Surface Scattering: Cancellations, Conditioning, and Convergence. J. Opt. Soc. Am. A, 21(4), 590-605.Google ScholarPubMed
[9] Nicholls, David P., and Reitich, Fernando. 2004b. Shape Deformations in Rough Surface Scattering: Improved Algorithms. J. Opt. Soc. Am. A, 21(4), 606-621.Google ScholarPubMed
[10] Nicholls, David P., and Nigam, Nilima. 2004. Exact Non-Reflecting Boundary Conditions on General Domains. J. Comput. Phys., 194(1), 278-303.CrossRefGoogle Scholar
[11] Nicholls, David P., and Nigam, Nilima. 2006. Error Analysis of a Coupled Finite Element/DtN Map Algorithm on General Domains. Numer. Math., 105(2), 267-298.Google Scholar
[12] Nicholls, David P., and Reitich, Fernando. 2005. On Analyticity of Traveling Water Waves. Proc. Roy. Soc. Lond., A, 461(2057), 1283-1309.CrossRefGoogle Scholar
[13] Nicholls, David P., and Reitich, Fernando. 2006. Rapid, Stable, High-Order Computation of Traveling Water Waves in Three Dimensions. Eur. J. Mech. B Fluids, 25(4), 406-424.CrossRefGoogle Scholar
[14] Akers, Benjamin F., and Nicholls, David P. 2010. Traveling Waves in Deep Water with Gravity and Surface Tension. SIAM J. App. Math., 70(7), 2373-2389.CrossRefGoogle Scholar
[15] Nicholls, David P. 2007. Spectral Stability of Traveling Water Waves: Analytic Dependence of the Spectrum. J. Nonlin. Sci., 17(4), 369-397.CrossRefGoogle Scholar
[16] Akers, Benjamin F., and Nicholls, David P. 2012b. Spectral Stability of Deep Two-Dimensional Gravity Water Waves: Repeated Eigenvalues. SIAM J. App. Math., 72(2), 689-711.CrossRefGoogle Scholar
[17] Akers, Benjamin F., and Nicholls, David P. 2012a. Spectral Stability of Deep Two-Dimensional Gravity-Capillary Water Waves. Stud. App. Math., 130, 81-107.Google Scholar
[18] Akers, Benjamin, and Nicholls, David P. 2014. Spectral Stability of Finite Depth Water Waves. Euro. J. Mech. B/Fluids, 46, 181-189.CrossRefGoogle Scholar
[19] Hu, Bei, and Nicholls, David P. 2005. Analyticity of Dirichlet-Neumann Operators on Holder and Lipschitz Domains. SIAM J. Math. Anal., 37(1), 302-320.CrossRefGoogle Scholar
[20] Hu|Bei, and Nicholls, David P. 2010. The Domain of Analyticity of Dirichlet-Neumann Operators. Proc. Royal Soc. Edinburgh A, 140(2), 367-389.Google Scholar
[21] Nicholls, David P., and Taber, Mark. 2008. Joint Analyticity and Analytic Continuation for Dirichlet-Neumann Operators on Doubly Perturbed Domains. J. Math. Fluid Mech., 10(2), 238-271.CrossRefGoogle Scholar
[22] Nicholls, David P., and Taber, Mark. 2009. Detection of Ocean Bathymetry from Surface Wave Measurements. Euro. J. Mech. B/Fluids, 28(2), 224-233.CrossRefGoogle Scholar
[23] Fazioli, C., and Nicholls, David P. 2008. Parametric Analyticity of Functional Variations of Dirichlet-Neumann Operators. Diff. Integ. Eqns., 21(5-6), 541-574.Google Scholar
[24] Fazioli, Carlo, and Nicholls, David P. 2010. Stable Computation of Variations of Dirichlet-Neumann Operators. J. Comp. Phys., 229(3), 906-920.CrossRefGoogle Scholar
[25] Nicholls, David P., and Shen, Jie. 2009. A Rigorous Numerical Analysis of the Transformed Field Expansion Method. SIAM J. Num. Anal., 47(4), 2708-2734.CrossRefGoogle Scholar
[26] Ladyzhenskaya, Olga A., and Ural'tseva, Nina N. 1968. Linear and quasilinear elliptic equations.New York: Academic Press.Google Scholar
[27] Friedman, Avner, and Reitich, Fernando. 2001. Symmetry-Breaking Bifurcation of Analytic Solutions to Free Boundary Problems: An Application to a Model of Tumor Growth. Trans. Amer. Math. Soc., 353, 1587-1634.CrossRefGoogle Scholar
[28] Chandezon, J., Maystre, D., and Raoult, G. 1980. A New Theoretical Method for Diffraction Gratings and its Numerical Application. J. Opt., 11(7), 235-241.CrossRefGoogle Scholar
[29] Chandezon, J., Dupuis, M.T., Cornet, G., and Maystre, D. 1982. Multicoated Gratings: A Differential Formalism Applicable in the Entire Optical Region. J. Opt. Soc. Amer., 72(7), 839.CrossRefGoogle Scholar
[30] Li, L., Chandezon, J., Granet, G., and Plumey, J. P. 1999. Rigorous and Efficient Grating-Analysis Method Made Easy for Optical Engineers. Appl. Opt., 38(2), 304-313.CrossRefGoogle ScholarPubMed
[31] Phillips, N. A. 1957. A Coordinate System Having Some Special Advantages for Numerical Forecasting. J. Atmos. Sci., 14(2), 184-185.Google Scholar
[32] Johnson, Claes, and Nédélec, J.-Claude. 1980. On the Coupling of Boundary Integral and Finite Element Methods. Math. Comp., 35(152), 1063-1079.CrossRefGoogle Scholar
[33] Han, Hou De, and Wu, Xiao Nan. 1985. Approximation of Infinite Boundary Condition and its Application to Finite Element Methods. J. Comput. Math., 3(2), 179-192.Google Scholar
[34] Keller, Joseph B., and Givoli, Dan. 1989. Exact Nonreflecting Boundary Conditions. J. Comput. Phys., 82(1), 172-192.CrossRefGoogle Scholar
[35] Givoli, Dan. 1991. Nonreflecting Boundary Conditions. J. Comput. Phys., 94(1), 1-29.CrossRefGoogle Scholar
[36] Givoli, Dan, and Keller, Joseph B. 1994. Special Finite Elements for Use with High-Order Boundary Conditions. Comput. Methods Appl. Mech. Engrg., 119(3-4), 199-213.CrossRefGoogle Scholar
[37] Givoli, Dan. 1992. Numerical methods for problems in infinite domains.Studies in Applied Mechanics, vol. 33. Amsterdam: Elsevier Scientific Publishing Co.CrossRefGoogle Scholar
[38] Grote, Marcus J., and Keller, Joseph B. 1995. On Nonreflecting Boundary Conditions. J. Comput. Phys., 122(2), 231-243.CrossRefGoogle Scholar
[39] Givoli, D. 1999. Recent Advances in the DtN FE Method. Arch. Comput. Methods Engrg., 6(2), 71-116.CrossRefGoogle Scholar
[40] Evans, Lawrence C. 1998. Partial differential equations.Providence, RI: American Mathematical Society.Google Scholar
[41] Gottlieb, David, and Orszag, Steven A. 1977. Numerical analysis of spectral methods: theory and applications.Philadelphia, PA.: Society for Industrial and Applied Mathematics. CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26.CrossRefGoogle Scholar
[42] Canuto, Claudio, Hussaini, M. Yousuff, Quarteroni, Alfio, and Zang, Thomas A. 1988. Spectral methods in fluid dynamics.New York: Springer-Verlag.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×