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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

[1] Strikwerda, John C. 2004. Finite difference schemes and partial differential equations. Second edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
[2] Morton, K. W., and Mayers, D. R 2005. Numerical solution of partial differential equations. Second edn. Cambridge University Press, Cambridge. An introduction.
[3] LeVeque, Randall J. 2007. Finite difference methods for ordinary and partial differential equations.Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Steady-state and time-dependent problems.CrossRefGoogle Scholar
[4] Thomas, J. W. 1995. Numerical partial differential equations: finite difference methods.Texts in Applied Mathematics, vol. 22. Springer-Verlag, New York.CrossRefGoogle Scholar
[5] Johnson, Claes. 1987. Numerical solution of partial differential equations by the finite element method.Cambridge: Cambridge University Press.Google Scholar
[6] Karniadakis, George Em, and Sherwin, Spencer J. 1999. Spectral/hp element methods for CFD.Numerical Mathematics and Scientific Computation. New York: Oxford University Press.Google Scholar
[7] Braess, Dietrich. 2001. Finite elements. Second edn. Cambridge: Cambridge University Press. Theory, fast solvers, and applications in solid mechanics, Translated from the 1992 German edition by Larry L., Schumaker.Google Scholar
[8] Hesthaven, Jan S., and Warburton, Tim. 2008. Nodal discontinuous Galerkin methods.Texts in Applied Mathematics, vol. 54. New York: Springer. Algorithms, analysis, and applications.CrossRefGoogle Scholar
[9] 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
[10] Canuto, Claudio, Hussaini, M. Yousuff, Quarteroni, Alfio, and Zang, Thomas A. 1988. Spectral methods in fluid dynamics.New York: Springer-Verlag.CrossRefGoogle Scholar
[11] Fornberg, Bengt. 1996. A practical guide to pseudospectral methods.Cambridge Monographs on Applied and Computational Mathematics, vol. 1. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
[12] Boyd, John P. 2001. Chebyshev and Fourier spectral methods. Second edn. Mineola, NY: Dover Publications Inc.Google Scholar
[13] Deville, M. O., Fischer, P. F., and Mund, E. H. 2002. High-order methods for incompressible fluid flow.Cambridge Monographs on Applied and Computational Mathematics, vol. 9. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
[14] Hesthaven, Jan S., Gottlieb, Sigal, and Gottlieb, David. 2007. Spectral methods for time-dependent problems.Cambridge Monographs on Applied and Computational Mathematics, vol. 21. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
[15] Colton, David, and Kress, Rainer. 1998. Inverse acoustic and electromagnetic scattering theory. Second edn. Berlin: Springer-Verlag.CrossRefGoogle Scholar
[16] Kress, Rainer. 1999. Linear integral equations. Second edn. New York: Springer-Verlag.CrossRefGoogle Scholar
[17] Bruno, Oscar P., and Reitich, Fernando. 1992. Solution of a boundary value problem for the Helmholtz equation via variation of the boundary into the complex domain. Proc. Roy. Soc. Edinburgh Sect. A, 122(3-4), 317-340.CrossRefGoogle Scholar
[18] Bruno, Oscar P., and Reitich, Fernando. 1993a. Numerical solution of diffraction problems: A method of variation of boundaries. J. Opt. Soc. Am. A, 10(6), 1168-1175.Google Scholar
[19] Bruno, Oscar P., and Reitich, Fernando. 1993b. Numerical solution of diffraction problems: A method of variation of boundaries. II. Finitely conducting gratings, Padé approximants, and singularities. J. Opt. Soc. Am. A, 10(11), 2307-2316.Google Scholar
[20] Bruno, Oscar P., and Reitich, Fernando. 1993c. Numerical solution of diffraction problems: A method of variation of boundaries. III. Doubly periodic gratings. J. Opt. Soc. Am. A, 10(12), 2551-2562.Google Scholar
[21] Bruno, Oscar P., and Reitich, Fernando. 1994. Approximation of analytic functions: A method of enhanced convergence. Math. Comp., 63(207), 195-213.CrossRefGoogle Scholar
[22] Bruno, Oscar P., and Reitich, Fernando. 1996. Calculation of electromagnetic scattering via boundary variations and analytic continuation. Appl. Comput. Electromagn. Soc. J., 11(1), 17-31.Google Scholar
[23] Bruno, Oscar P., and Reitich, Fernando. 1998. Boundary-variation solutions for bounded-obstacle scattering problems in three dimensions. J. Acoust. Soc. Am., 104(5), 2579-2583.CrossRefGoogle Scholar
[24] Bruno, Oscar P., and Reitich, Fernando. 2001. High-order boundary perturbation methods. Pages 71-109 of: Mathematical Modeling in Optical Science, vol. 22. Philadelphia, PA: SIAM. Frontiers in Applied Mathematics Series.Google Scholar
[25] Milder, D. Michael. 1991a. An improved formalism for rough-surface scattering of acoustic and electromagnetic waves. Pages 213-221 of: Proceedings ofSPIE - The International Society for Optical Engineering (San Diego, 1991), vol. 1558. Bellingham, WA: Int. Soc. for Optical Engineering.Google Scholar
[26] Milder, D. Michael. 1991b. An improved formalism for wave scattering from rough surfaces. J. Acoust. Soc. Am., 89(2), 529-541.CrossRefGoogle Scholar
[27] Milder, D. Michael, and Sharp, H. Thomas. 1991. Efficient computation of rough surface scattering. Pages 314–322 of: Mathematical and numerical aspects of wave propagation phenomena (Strasbourg, 1991).Philadelphia, PA: SIAM.Google Scholar
[28] Milder, D. Michael, and Sharp, H. Thomas. 1992. An improved formalism for rough surface scattering. II: Numerical trials in three dimensions. J. Acoust. Soc. Am., 91(5), 2620-2626.CrossRefGoogle Scholar
[29] Craig, Walter, and Sulem, Catherine. 1993. Numerical simulation of gravity waves. J. Comput. Phys., 108, 73-83.CrossRefGoogle Scholar
[30] Milder, D. Michael. 1996b. Role of the admittance operator in rough-surface scattering. J. Acoust. Soc. Am., 100(2), 759-768.CrossRefGoogle Scholar
[31] Milder, D. Michael. 1996a. An improved formalism for electromagnetic scattering from a perfectly conducting rough surface. Radio Science, 31(6), 1369-1376.CrossRefGoogle Scholar
[32] 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
[33] 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
[34] Nicholls, David P., and Reitich, Fernando. 2003. Analytic continuation of Dirichlet-Neumann operators. Numer. Math., 94(1), 107-146.CrossRefGoogle Scholar
[35] Lamb, Horace. 1993. Hydrodynamics. sixth edn. Cambridge: Cambridge University Press.Google Scholar
[36] Zakharov, Vladimir. 1968. Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. App. Mech. Tech. Phys., 9, 190-194.Google Scholar
[37] Rayleigh, Lord. 1907. On the dynamical theory of gratings. Proc. Roy. Soc. London, A79, 399-416.CrossRefGoogle Scholar
[38] Rice, S. O. 1951. Reflection of electromagnetic waves from slightly rough surfaces. Comm. Pure Appl. Math., 4, 351-378.CrossRefGoogle Scholar
[39] 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
[40] 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
[41] Malcolm, Alison, and Nicholls, David P. 2011. A field expansions method for scattering by periodic multilayered media. J. Acoust. Soc. Am., 129(4), 1783-1793.CrossRefGoogle ScholarPubMed
[42] Fang, Zheng, and Nicholls, David P. 2014. An operator expansions method for computing Dirichlet-Neumann operators in linear elastodynamics. J. Comput. Phys., 272, 266-278.CrossRefGoogle Scholar
[43] Nicholls, David P. 2015. A method of field expansions for vector electromagnetic scattering by layered periodic crossed gratings. J. Opt. Soc. Am., A (to appear).
[44] Baker, Jr., George A., and Graves-Morris, Peter. 1996. Padé approximants. Second edn. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
[45] Bender, Carl M., and Orszag, Steven A. 1978. Advanced mathematical methods for scientists and engineers.New York: McGraw-Hill Book Co. International Series in Pure and Applied Mathematics.Google Scholar

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