Spectral Methods: Fundamentals in Single Domains

584 Pages·2006·10.17 MB·English

by  Claudio Canuto, M. Youssuff Hussaini, Alfio Quarteroni, Thomas A. Zang (auth.)

Scientific Computation
EditorialBoard
J.-J.Chattot,Davis,CA,USA
P.Colella,Berkeley,CA,USA
WeinanE,Princeton,NJ,USA
R.Glowinski,Houston,TX,USA
M.Holt,Berkeley,CA,USA
Y.Hussaini,Tallahassee,FL,USA
P.Joly,LeChesnay,France
H.B.Keller,Pasadena,CA,USA
J.E.Marsden,Pasadena,CA,USA
D.I.Meiron,Pasadena,CA,USA
O.Pironneau,Paris,France
A.Quarteroni,Lausanne,Switzerland
andPolitecnicoofMilan,Italy
J.Rappaz,Lausanne,Switzerland
R.Rosner,Chicago,IL,USA
P.Sagaut,Paris,France
J.H.Seinfeld,Pasadena,CA,USA
A.Szepessy,Stockholm,Sweden
M.F.Wheeler,Austin,TX,USA
C. Canuto M. Y. Hussaini
A. Quarteroni T. A. Zang
Spectral Methods
Fundamentals in Single Domains
With106Figuresand19Tables
123
ClaudioCanuto M.YousuffHussaini
DipartimentodiMatematica SchoolofComputationalScience
PolitecnicodiTorino FloridaStateUniversity
CorsoDucadegliAbruzzi,24 Tallahassee,FL32306-4120,USA
10129Torino,Italy e-mail:[email protected]
e-mail:[email protected]
AlfioQuarteroni ThomasA.Zang,Jr.
SB-IACS-CMCS,EPFL NASALangleyResearchCenter*
1015Lausanne,Switzerland MailStop449
and Hampton,VA23681-2199,USA
MOX,PolitecnicodiMilano e-mail:[email protected]
PiazzaLeonardodaVinci,32
20133Milano,Italy
e-mail:alfio.quarteroni@epfl.ch
*ThisdoesnotconstituteanendorsementofthisworkbyeithertheU.S.GovernmentortheNASA
LangleyResearchCenter.
Coverpicture:SeeFig.4.4(left)
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Preface
As a tool for large-scale computations in fluid dynamics, spectral methods
were originally proposed in 1944 by Blinova, first implemented in 1954 by
Silberman, virtually abandoned in the mid-1960s, resurrected in 1969–70 by
Orszag and by Eliason, Machenhauer and Rasmussen, developed for special-
ized applications in the 1970s, endowed with the first mathematical foun-
dations by the seminal work of Gottlieb and Orszag in 1977, extended to
a broader class of problems and thoroughly analyzed in the 1980s, and en-
tered the mainstream of scientific computation in the 1990s. Two decades
ago when we wrote Spectral Methods in Fluid Dynamics (1988) both the
subject and the authors were barely past their adolescence. As the field and
the authors are now in their middle age, the time seems ripe for a more
mature discussion of the field, accounting for the main contributions of the
intervening years. Motivated by the many favorable comments we have re-
ceivedand the continuing interest in the first book (which will be referred to
asCHQZ1),yetdesiringtopresentamoremodernperspective,weembarked
on a project which has resulted in this book (referred to as CHQZ2) and
itscompanionbook(Canuto,Hussaini,QuarteroniandZang(2007),referred
to as CHQZ3). These, like our first text on this subject, are books about
spectral methods for partial differential equations – when to use them, how
to implement them, and what can be learned from their rigorous theory.
The original promoters of spectral methods were meteorologists study-
ing global weather modeling and fluid dynamicists investigating isotropic
turbulence. The converts who were inspired by the successes of these pio-
neersremained,forthemostpart,confinedtotheseandcloselyrelatedfields
throughout the 1970s. During that decade spectral methods appeared to be
well-suited only for problems governed by ordinary differential equations or
by partial differential equations with (mostly) periodic boundary conditions.
And, of course, the solution itself needed to be smooth.
Both the theory and the algorithms of classical (single-domain) spectral
methods for smooth problems were already reasonably mature in the mid-
1980s. On the theoretical side, approximation theory results were available
for periodic and nonperiodic problems, stability and convergence analyses
were in-hand for steady and unsteady linear problems, and detailed numer-
ical analyses had been produced for a variety of methods for fluid dynam-
VI Preface
ics applications, and particularly for the incompressible Navier-Stokes equa-
tions. Open issues included discontinuous problems (with compressible flows
of particular interest), convergence analysis of iterative methods, artificial
outflow boundary conditions, and rigorous analysis of time discretizations.
On the algorithms front, explicit methods for fully periodic problems were
routine,efficientdirectsolutionmethodswereavailableforseveralimportant
constant-coefficient implicit equations, numerous efficient algorithms were
available for incompressible flows with at most one nonperiodic direction,
and shock-fitting methods had been developed for compressible flows. Nu-
merous approaches were being tried for discontinuous problems, especially
for shock capturing in compressible flows. Rapid developments were taking
place in iterative methods for implicit equations. The extension of spectral
methods to problems in complex geometries through multidomain spectral
approaches was proceeding explosively.
Singular progress has indeed been made over the past two decades in ex-
tendingspectralmethodstoarbitrarygeometries,enablingwhatsomewould
considerthemathematicalnirvanaofamethodofarbitrarilyhighordercap-
ableofapplicationtoproblemsonanarbitrarygeometry.Inthisrespect,the
trajectory of spectral methods over the past 20 years has been converging
towards that of hp finite-element methods.
This process of migration from single-domain to multidomain spectral
methods has required the injection of novel mathematical tools, and stimu-
lated original investigation directions. Mathematics has had a profound im-
pact on the correct design and interpretation of the methods, and in some
casesithasinspiredthedevelopmentofdiscontinuousspectralmethods(such
asthemortarmethodandthediscontinuousGalerkinmethod)evenforprob-
lemswithcontinuoussolutions.Ontheotherhand,sinceingeneralageomet-
rically complex computational domain is split into polygonal or polyhedral
subdomains (or elements), tensor-product domains are no longer a prerequi-
siteforspectralmethods,withthedevelopmentofspectralbasesontriangles
and tetrahedra.
Oneofthemostpronouncedchangesisthatthestrongformofdifferential
equationshaslostitsprimacyastheanchorforthediscretizationoftheprob-
lem.Multidomain spectralmethodsaremoreeasilyandreliablyapproached,
both algorithmically and theoretically, from weak formulations of the differ-
ential equations. Moreover, the use of many subdomains has motivated the
use of moderate polynomial degrees in every subdomain – small from the
perspective of classical spectral methods, but large from the perspectives of
finite-difference and finite-element methods. From a theoretical viewpoint,
new error estimates have been established for which the roles of the local
polynomial degree and the geometrical size of the local elements are both
captured. From an algorithmic point of view, the role of matrices has been
addressedingreatdetail,correspondingtotheincreasedinterestinsmalland
moderate values of N and on techniques of matrix assembly. Exploitation of
Preface VII
advanced linear algebra tools for sparse, ill-conditioned systems has become
of paramount importance.
Inspiteofthismajorchangeofperspective,thenewmultidomainspectral
methodsstillenjoysomeofthemostdistinguishing(anddesirable)featuresof
“classical”spectralmethods–Gaussianintegrationformulas,lowdispersion,
and ease of preconditioning by low-order discretization matrices.
Over the past twenty years the appeal of spectral methods for applica-
tions such as computational fluid dynamics has expanded, commensurate
with the erosion of most of the obstacles to their wider application. Beyond
the specific techniques, the culture of high-order methods has entered the
background knowledge of numerical analysts. Spectral methods have been
traditional in academic instruction since the 1990s and began to penetrate
industrial applications this decade. In fact, spectral methods are successfully
usednowadaysforwidelydiverseapplications,suchaswavepropagation(for
acoustic, elastic, seismic and electromagnetic waves), solid and structural
analysis, marine engineering, biomechanics, astrophysics, and even financial
engineering.Theirprincipalappealintheacademicresearchenvironmentstill
reliesontheirsuperiorrateofconvergence,whichmakesthemanidealvirtual
lab.Intheindustrial(extra-academic)environment,spectral-basedcodesare
appreciated, andoften preferred,owingtothe low dissipation and dispersion
errors,theneatwaytotreatboundaryconditions,and,today,theavailability
ofefficientalgebraicsolversthatallowafavorabletrade-offbetweenaccuracy
and computational cost.
The basics of classical spectral methods remain essential for current re-
search on the frontiers of both the algorithms and the theory of spectral
methods.Atthesametime,multidomainspectralmethodshavealreadywar-
rantedbooksintheirownright.Ourobjectiveswiththecurrenttwobooksare
to modernize our thorough discussion of classical spectral methods, account-
ing for advances in the theory and more extensive application experience in
the fluid dynamics arena, while summarizing the current state of multido-
main spectral methods from the perspective of classical spectral methods.
Themajormethodologicaldevelopmentsinclassicalspectralmethodsduring
the past two decades have been the emergence of the Galerkin with numer-
ical integration (G-NI) approach, the decline of the tau method to a niche
role,improvedtreatmentofboundaryconditions,theadaptationofadvanced
direct and iterative methods to spectral discretizations also thanks to a bet-
ter insight into the mathematical basis of preconditioning, the development
of more sophisticated tools to control spurious high-frequency oscillations
without losing the formal accuracy of the method, and the formulation of
spectral discretizations on triangles (in two dimensions) and tetrahedra (in
three dimensions). From the applications perspective in fluid dynamics, new
algorithms have been produced for compressible linear and secondary stabil-
ity, for parabolized stability equations, for velocity-vorticity formulations of
incompressible flow,and forlarge-eddysimulations, alongwith refinementof
VIII Preface
spectral shock-fitting methods. Moreover, the once intense debate over the
impact of aliasing errors has settled down to polite differences of opinion.
While asignificant amountofmaterialinthetwonewbooks has beenre-
tainedfromportionsofourearliertext,CHQZ1,themajorityofthematerial
is new. The most consistent augmentation is that all chapters are enhanced
by the addition of material for the G-NI method. The added material has
necessitated publishing this new work as two separate books. The rationale
for the division of the material between the two books is that we furnish in
this first book, CHQZ2, a comprehensive discussion of the generic aspects
of classical spectral methods, while the second book, CHQZ3, focuses on
applications to fluid dynamics and on multidomain spectral methods.
Chapters 1–4 of the present book are of general interest. Chapter 1 pro-
vides a motivational introduction to spectral methods, as well as a preview
of the more sophisticated single-domain applications in fluid dynamics pre-
sentedinthesecondbook.Chapter2containsathoroughdiscussionofclassi-
cal orthogonal expansions, supplemented with a basic description of spectral
approximationsontrianglesandtetrahedra.Chapter3providesacomprehen-
siveguidetospectraldiscretizationsinspaceforpartialdifferentialequations
in one space dimension, using the Burgers equation model problem for illus-
trative purposes. A discussion of boundary conditions for hyperbolic equa-
tions, and detailed prescriptions for the construction of mass and stiffness
matrices for elliptic problems are also given. Chapter 4 focuses on solution
techniques for the algebraic systems generated by spectral methods. In ad-
dition to a number of now classical results, the chapter offers a thorough
investigation of modern direct and iterative methods, as befits the extensive
developments that have transpired in the past two decades. A large number
of original numerical examples are presented in these two chapters. Chap-
ters 5–7 focus on the mathematical theory of classical spectral methods.
Chapter 5 consists of a review of those results from approximation theory
which are pertinent to the theoretical analysis of spectral methods. Most of
them are classical; however a few of them are newer, as they highlight the
dependence on both polynomial degree and geometrical parameters for both
tensor-product domains and simplicial domains (triangles and tetrahedra).
Chapter 6 is the focal point of this book regarding the theory of spectral
methods. The fundamental stability and convergence results are established
forallkindsofnumericalspectralapproximations(Galerkin,tau,collocation,
and G-NI) to linear partial differential equations, both steady and unsteady.
Finally, Chap. 7 addresses the theoretical analysis of spectral approxima-
tions to a family of partial differential equations that can be regarded as
the building blocks of mathematical modelling in continuum mechanics in
general, and in fluid dynamics in particular. It places particular emphasis
on the Poisson equation, singularly perturbed elliptic equations that govern
advection-diffusion and reaction-diffusion processes featuring sharp bound-
ary layers, the heat equation, hyperbolic equations and systems, and the
Preface IX
steady Burgers equation. Moreover, it addresses the eigenvalue analysis of
matrices produced by spectral approximations, and illustrates recent tech-
niques to resolve the Gibbs phenomenon for discontinuous solutions through
filtering, singularity detection and spectral reconstruction techniques. The
first book ends with four Appendices surveying several algorithmic and the-
oretical numerical analysis topics that are not specific to spectral methods,
but of sufficient utility to some readers to warrant inclusion. In Appendix
A we review some basic notations and theorems from functional analysis.
Appendix B reviews the fast Fourier transform and some adaptations that
are particularly useful to Fourier and Chebyshev methods. Appendix C is
a gentle introduction to iterative methods and lists several specific iterative
algorithms that have been exploited in spectral methods, while Appendix D
describes some basic concepts, specific numerical schemes, and stability re-
gionsforthosetemporaldiscretizationsthathavebeenfavoredbythespectral
methods community.
In our second book (Canuto, Hussaini, Quarteroni and Zang (2007)),
Chap. 1 covers the basic equations of fluid mechanics. Chapter 2 is solely
devoted to spectral algorithms for analyses of linear and nonlinear stability
offluid flows.Applications tocompressible flows andtoparabolized stability
equations post-date our earlier book. Chapter 3, on algorithms for incom-
pressible flows, has a sharp emphasis on those algorithms that remained in
reasonablyextensiveusepost-1990andprovidesamoderndiscussionofsolu-
tion techniques for problems with twoormorenonperiodic directions. Chap-
ter 4, on algorithms for hyperbolic systems and compressible flows, empha-
sizes algorithms for enforcing boundary conditions, methods for computing
homogeneous,compressibleflows,andanimprovedapproachtoshockfitting.
Chapter 5 introduces the main strategies to construct spectral approxima-
tionsincomplexdomains,andinparticularthespectral-elementmethod,the
mortar-elementmethod,thespectraldiscontinuous Galerkinmethod,aswell
as the more traditional patching collocation method. Their theoretical prop-
erties are analyzed, and their algebraic aspects are investigated. Chapter 6
illustrates solution strategies based on domain decomposition techniques for
the spectral discretizations investigated in Chap. 5. Both Schur-based and
Schwarz-basediterativesolversandpreconditionersareconsidered,andtheir
computationaladvantages(inparticular,theirpropertyofscalabilitywithre-
specttothenumberofsubdomains) areillustrated.Ourprojectclosesinthe
same manner in which it began, with a survey of representative large-scale
applications of (this time multidomain) spectral methods.
Whereas with our first text we made a valiant effort to provide compre-
hensivecoverageofallavailablespectralmethods(atleastforfluiddynamics
applications) and to provide a bibliography that encompassed all extant ref-
erences to spectral methods, here we acknowledge the practical impossibility
ofsuchanambitioninthefaceofalltheworkthathassincetranspiredinthe
field.Westillaimtoprovidecomprehensivecoverageofgeneralmethodology.
X Preface
However, our coverage of particular algorithms is necessarily representative
rather than complete. Our aim is to focus on those algorithms that have
stood the test of time in fluid dynamical applications, as assessed by how
widely they have been used in the past two decades. But our knowledge in
this area is certainly not exhaustive, and others would no doubt have made
somewhatdifferentchoices.Inourcitationsweenforceastrongpreferencefor
archivalpublications.Werecognizethatmanydevelopmentsappearedearlier
(in some cases many years earlier) in pre-prints or conference publications.
Butweonly citenon-archival sourceswhennoarchivalreferenceis available.
Themanynumericalexamplesproducedexpresslyforthesebookshaveall
been run on desktop computers (under both Linux and Macintosh operating
systems), usually in 64-bit arithmetic with the standard IEEE precision of
2−52 ≈2×10−16.Ahalf-dozenorsodifferentcomputerswereemployed,with
clockspeedsontheorderof1–3GHz;someofthesecomputershadtwoCPUs.
The workhorse languages were Matlab and Fortran, with no special effort
devotedtofine-tuningtheperformanceofthecodes.Thereaderwillcertainly
appreciate that the occasional timings presented here are meant solely to
provide a rough comparison between the costs of alternative algorithms and
should not be construed as representing a definitive verdict on the efficiency
of the methods.
Nowadays, considerable software for spectral methods is freely available
on the web, ranging from libraries of basic spectral operations all the way to
complete spectral codes for Navier-Stokes (and other complex) applications.
Due to the highly dynamic nature of these postings, we have chosen not to
list them in the text (except to acknowledge codes that we have used here
for numerical examples), but to maintain a reasonably current list of such
sources on the Web site (http://www.dimat.polito.it/chqz/) for this and the
companion text. There is always the possibility that this site itself may need
tobemovedduetounforeseencircumstances;inthateventoneshouldcheck
the Springer site for the link to the detailed book Web site.
The authors are grateful to Dr. Wolf Beiglbo¨ck, Dr. Ramon Khanna
and the Springer staff for their patience while waiting for our long overdue
manuscript. The authors are pleased to acknowledge the many discussions
and helpful comments on the manuscript that have been provided by col-
leagues such as Paola Gervasio, David Kopriva, Giovanni Monegato, Luca
Pavarino and Andrea Toselli. The technical support of Paola Gervasio and
Marco Discacciati in running numerical tests, preparing figures and tables,
typing and editing a significant part of the whole manuscript is gratefully
acknowledged. Thanks are also due to Stefano Berrone and Sophie Fosson
for providing further technical support, and to Susan Greenwalt for her ad-
ministrative support of this project. We appreciate the generosity of those
individuals who have given us permission to reprint figures from their work
in these texts. The authors are grateful to the Politecnico di Torino, the
Florida State University, the Ecole Polytechnique F´ed´erale de Lausanne and
Preface XI
the Politecnico di Milano for their facilitation of this endeavor. One of us
(MYH) is particularly grateful to Provost Lawrence Abele of Florida State
Universityforhisencouragementandsupportforthisproject.Finally,weare
most appreciative of the support and understanding we have received from
our wives (Manuelita, Khamar, Fulvia and Ann) and children (Arianna, Su-
sanna, Moin, Nadia, Marzia and Silvia) during this project.
Torino, Italy Claudio Canuto
Tallahassee, Florida M. Yousuff Hussaini
Lausanne, Switzerland and Milano, Italy Alfio Quarteroni
Carrollton, Virginia Thomas A. Zang
February, 2006