Table Of ContentScientific Computation
EditorialBoard
J.-J.Chattot,Davis,CA,USA
P.Colella,Berkeley,CA,USA
W.E,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
Evolution to Complex Geometries
and Applications to Fluid Dynamics
WithFiguresandTables
123
ClaudioCanuto M.YousuffHussaini
DipartimentodiMatematica SchoolofComputationalScience
PolitecnicodiTorino FloridaStateUniversity
CorsoDucadegliAbruzzi, Tallahassee,FL-,USA
Torino,Italy e-mail:myh@scs.fsu.edu
e-mail:claudio.canuto@polito.it
AlfioQuarteroni ThomasA.Zang,Jr.
SB-IACS-CMCS,EPFL NASALangleyResearchCenter*
Station MailStop
Lausanne,Switzerland Hampton,VA-,USA
and e-mail:Thomas.A.Zang@nasa.gov
MOX,PolitecnicodiMilano
PiazzaLeonardodaVinci,
Milano,Italy
e-mail:alfio.quarteroni@epfl.ch
* This does not constitute an endorsement of this work by either the U.S. Government or the NASA
LangleyResearchCenter.
Coverfigures:Upperleft:StructuraldynamicsanalysisoftheRomanColosseum:spectralelementcalcula-
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anaircraftsurface(Fig..).Lowerleft:VelocitycontoursforaTaylor-Greenvortexsimulationusing
aFourierspectralmethod(Fig..).Lowerright:Spectralelementapproximationofaneigenfunctionof
theLaplaceoperatorwithDirichletboundaryconditionsinasquare(Fig..).
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Preface
TwodecadesagowhenwewroteSpectral Methods in Fluid Dynamics (1988),
thesubjectwasstillfairlynovel.Motivatedbythemanyfavorablecomments
we have received and the continuing interest in that book (which will be
referred to as CHQZ1), and yet desiring to present a more modern perspec-
tive, we embarkedon the project which resulted in our recent book (Canuto
et al. (2006), referred to as CHQZ2) and the present new book (referred to
as CHQZ3). Our objectives with these two new books are to modernize our
thoroughdiscussionofclassicalspectralmethods,accountingforadvancesin
the theory and more extensive application experience in the fluid dynamics
arena,while summarizingthe currentstate ofmultidomainspectralmethods
from the perspective of classicalspectral methods. While the two new books
drawextensivelyfromportionsofourearliertext,CHQZ1,muchofCHQZ2,
and most of CHQZ3 is new. The added content has necessitated our pub-
lishing this new work as two separate books. The rationale for the division
of the material between the books is that we furnished in the first new book
a comprehensive discussion of the fundamental aspects of classical spectral
methods in single domains. This second new book focuses on applications to
fluid dynamics and on multidomain spectral methods.
The historical evolution of spectral methods from their initial (now clas-
sical) versions to the contemporary multidomain versions has been covered
in some detail in the Preface of CHQZ2. In short, both the theory and the
algorithmsofclassicalspectralmethodsforsmoothproblemswerereasonably
mature already in the mid-1980s, and singular progress has been made over
the past two decades in extending spectral methods to arbitrarygeometries,
enabling what some would consider the mathematical nirvana of a method
of arbitrarily high-order capable of application to problems on an arbitrary
geometry. In this respect, the trajectory of spectral methods over the past
20 years has been approaching that of hp finite-element methods. This pro-
cess of migration from single-domain to multidomain spectral methods has
requiredtheinjectionofnovelmathematicaltoolsandstimulatedoriginalin-
vestigationdirections.Mathematicshashadaprofoundimpactonthecorrect
design and interpretation of the methods, and in some cases it has inspired
the development of discontinuous spectral methods even for problems with
continuous solutions. On the other hand, since in general a geometrically
VI Preface
complex computational domain is split into polygonal or polyhedral subdo-
mains (or elements), tensor-productdomains are no longera prerequisite for
spectral methods, with the development of spectral bases on such elements
as triangles and tetrahedra. Moreover, the splitting into many subdomains
hasmotivatedtheuseofmoderatepolynomialdegreesineverysubdomain—
small from the perspective of classical spectral methods, but large from the
perspectivesoffinite-volumeandfinite-elementmethods.Inspite ofthis ma-
jor change of perspective, the new multidomain spectral methods still enjoy
some of the most distinguishing (and desirable) features of “classical” spec-
tral methods—Gaussian integration formulas, low dispersion, and ease of
preconditioning by low-order discretization matrices.
The main contents of the current text are as follows: Chapter 1 surveys
the relevantequations offluid dynamics,including the historicalbackground
and the underlying assumptions of the mathematical models. Chapter 2
presentsspectralalgorithmsforsolutionoftheboundary-layerequationsand
for analyses of linear and nonlinear stability of fluid flows. Chapter 3, on
single-domain algorithms for incompressible flows, focuses its discussion of
spectral methods for problems with at most one nonperiodic direction on
those algorithms that remained in reasonably extensive use post-1990. Fur-
thermore, it surveys the basic spatial discretization schemes for problems
with two or more nonperiodic directions; these schemes are the foundation
of many multidomain spectral methods for incompressible flows. Chapter 4,
on classicalspectral methods for hyperbolic systems and compressible flows,
emphasizes algorithms for enforcing boundary conditions, methods for com-
puting smooth compressible flows, and spectrally accurate solution of dis-
continuous compressible flows using the shock-fitting approach. Chapter 5
introduces the main strategies for constructing spectral approximations in
complex domains, in particular, the spectral element method, the mortar el-
ement method, the spectral discontinuous Galerkin method, as well as the
more traditional patching collocation method. Their theoretical properties
are analyzed and their algebraic aspects are investigated. Some of the per-
spectives and results coincide with those of the hp version of finite-element
methods (apart from the difference in notation). We arrive at them from
the “spectralmethod”road,i.e.,beingcoherentwiththespectraltechnology
within each element while reducing the size of the elements (the alternative
roadstartsfromthe h versionofFEMandconservesthe typicalglobal/local
interplaywhile increasingthe polynomialdegreeineachelement).Chapter6
illustrates solution strategies based on domain decomposition techniques for
the spectral discretizations investigated in Chap. 5. Both Schur-based and
Schwarz-basediterativesolversandpreconditionersareconsidered,andtheir
computational advantages (in particular, their property of scalability with
respect to the number of subdomains) are illustrated. Chapter 7 caps our
discussion of multidomain spectral methods with its survey of the modern
Preface VII
approachtocomputingincompressibleflowsingeneralgeometriesusinghigh-
order, spectral discretizations.
HavingCHQZ2athandisnotnecessaryforreadingCHQZ3,asallessen-
tial formulas are repeated here (in Chap. 8) in a short “primer for classical
spectralmethods”,andwehavealsoincludedthekeyelementsofAppendices
A–DfromCHQZ2.(Theseappendicescovermaterialon(A)somebasicnota-
tionsandtheoremsfromfunctionalanalysis,(B)theFastFourierTransform,
(C) iterative methods, and (D) temporal discretizations.) However, CHQZ2
does contain considerable material necessary for a better understanding of
the subject,andwe do makecopiousreferencesto specific andmoredetailed
material in CHQZ2. The complete Table of Contents from CHQZ2 is listed
rightafterthatofthepresentbook,asaconveniencetothosereaderswishing
to locate specific backgroundmaterial.Also,the newAppendix D.3 contains
some useful technical details for, but peripheral to, some of the algorithms
discussed in the text.
In contrast to other recent expositions on the subject, such as the books
by Karniadakis and Sherwin (2005) and Deville et al. (2002), this work cer-
tainly contains fewer implementation details of multidomain spectral algo-
rithms. On the other hand, we accompany the algorithmic description with
the theoretical framework necessary for the a priori prediction of their per-
formance,whichinturnsupportsinformedchoicesfortheiruseandprovides
somecriteriaforcodeverification.Wetrustthatourcopiousnumericalexam-
ples,andespeciallythe large-scaleapplications,provideconvincingempirical
evidence that spectral methods have indeed evolved sufficiently that even in
complex geometries they provide efficient, highly accurate numerical simula-
tions of diverse physical phenomena. We hope that this effort, together with
the other books and the ever-expanding literature on the subject, will foster
the wider penetration of higher-order ideas and methods in computational
engineering and science. The level of diffusion and appreciation that is by
now granted to higher-order methods in certain communities, such as the
structural mechanics one, is certainly a positive trend.
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 spectralmethods, here we acknowledgethe practicalimpossibility
ofsuchanambitioninthefaceofalltheworkthathassincetranspiredinthe
field.Westillaimtoprovidecomprehensivecoverageofgeneralmethodology.
However,ourdiscussionofparticularalgorithmsis necessarilyrepresentative
ratherthan complete.We focus onthose single-domainalgorithmsthathave
seen the widest use in fluid dynamics applications and on a broad represen-
tation of multidomain algorithms. However, our knowledge in this area is
certainly not exhaustive, and others would no doubt have made somewhat
different choices. In our citations we enforce a strong preference for archival
publications.Werecognizethatmanydevelopmentsappearedearlier(insome
VIII Preface
casesmanyyearsearlier)inpre-printsorconferencepublications,butweonly
cite non-archival sources when no archival reference is available.
Themanynumericalexamplesproducedexpresslyforthisbookhavebeen
run on desktop computers in 64-bit arithmetic unless otherwise noted.
Nowadays, considerable software for spectral methods is freely available
ontheWeb,rangingfromlibrariesofbasicspectraloperationsallthe wayto
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
companiontext. There is alwaysthe possibility that this site itself may need
to be moved due to unforeseen circumstances; in this case, one should check
the Springer site for the link to the detailed Web site of the book.
The authors are grateful to Dr. Wolf Beiglbo¨ck, Dr. Ramon Khanna and
the Springer staff for their encouragement and facilitation of this project.
ThetechnicalsupportofPaolaGervasioandMarcoDiscacciatiinperforming
numericaltests,preparingfiguresandtables,typing,andeditingmuchofthe
manuscript was indispensable. The authors are pleased to acknowledge the
many discussions and helpful comments on the manuscript that have been
providedbycolleaguessuchasErikBurman,PaolaGervasio,DavidKopriva,
MujeebMalik,RobertMoser,LucaPavarino,DavidPruett,JamieQuirk,and
AndreaToselli.WeappreciatetheassistanceprovidedbyDavidPruett,David
Kopriva,andBenjaminStammforseveralofthenumericalexamples.Thanks
are also due to Stefano Berrone, Dilek Dustegor, Giuseppe Ghib`o, Nicola
Parolini, and Svetlana Poroseva for providing additional technical support.
We are also indebted to the many individuals who have graciously given us
permissiontoreprintfiguresfromtheirwork.Theauthorsaregratefultothe
Politecnico di Torino, Florida State University (esp. Lawrence Abele), the
Ecole Polytechnique F´ed´erale de Lausanne, the Politecnico di Milano, and
the NASA Langley Research Center for their facilitation of this endeavor.
Finally, we are most appreciative of the support and understanding we have
receivedfrom our wives (Manuelita,Khamar,Fulvia, and Ann) and children
(Arianna, Susanna, Moin, Nadia, Marzia, and Silvia) during this project.
Torino, Italy Claudio Canuto
Tallahassee, Florida M. Yousuff Hussaini
Lausanne, Switzerland and Milan, Italy Alfio Quarteroni
Hampton, Virginia Thomas A. Zang
June, 2007
Contents
1. Fundamentals of Fluid Dynamics ......................... 1
1.1 Introduction ........................................... 1
1.2 Fluid Dynamics Background ............................. 1
1.2.1 Phases of Matter ................................. 2
1.2.2 Thermodynamic Relationships ..................... 3
1.2.3 Historical Perspective............................. 6
1.3 Compressible Fluid Dynamics Equations .................. 7
1.3.1 Compressible Navier–Stokes Equations .............. 8
1.3.2 Nondimensionalization ............................ 12
1.3.3 Navier–Stokes Equations with Turbulence Models .... 13
1.3.4 Compressible Euler Equations...................... 17
1.3.5 Compressible Potential Equation ................... 17
1.3.6 Compressible Boundary-LayerEquations ............ 19
1.3.7 Compressible Stokes Limit......................... 20
1.3.8 Low Mach Number Compressible Limit ............. 21
1.4 Incompressible Fluid Dynamics Equations ................. 21
1.4.1 Incompressible Navier–Stokes Equations............. 21
1.4.2 Incompressible Navier–Stokes Equations
with Turbulence Models........................... 22
1.4.3 Vorticity–Streamfunction Equations ................ 25
1.4.4 Vorticity–Velocity Equations....................... 26
1.4.5 Incompressible Boundary-LayerEquations........... 27
1.5 Linear Stability of ParallelFlows ......................... 27
1.5.1 Incompressible Linear Stability..................... 29
1.5.2 Compressible Linear Stability ...................... 31
1.6 Stability Equations for Nonparallel Flows.................. 36
2. Single-Domain Methods for Stability Analysis ............ 39
2.1 Introduction ........................................... 39
2.2 Boundary-LayerFlows .................................. 41
2.2.1 Incompressible Boundary-LayerFlows............... 41
2.2.2 Compressible Boundary-LayerFlows................ 48
2.3 Linear Stability of Incompressible ParallelFlows ........... 52
2.3.1 Spectral Approximations for Plane Poiseuille Flow.... 52
X Contents
2.3.2 Numerical Examples for Plane Poiseuille Flow ....... 57
2.3.3 Some Other Incompressible Linear Stability Problems. 61
2.4 Linear Stability of Compressible ParallelFlows............. 64
2.5 Nonparallel Linear Stability.............................. 69
2.5.1 Linear Parabolized Stability Equations.............. 69
2.5.2 Two-Dimensional Global Stability Analysis .......... 71
2.6 Transient Growth Analysis .............................. 72
2.7 Nonlinear Stability ..................................... 75
2.7.1 Quasi-Steady Finite-Amplitude Solutions............ 75
2.7.2 Secondary Instability Theory ...................... 77
2.7.3 Nonlinear Parabolized Stability Equations ........... 81
3. Single-Domain Methods for Incompressible Flows ........ 83
3.1 Introduction ........................................... 83
3.2 Conservation Properties and Time-Discretization ........... 86
3.2.1 Conservation Properties........................... 86
The Rotation Form ............................... 88
The Skew-Symmetric Form ........................ 90
Convection and Divergence Forms .................. 92
3.2.2 General Guidelines for Time-Discretization .......... 92
3.2.3 Coupled Methods ................................ 93
Fully Implicit Schemes ............................ 93
Semi-Implicit Schemes ............................ 93
3.2.4 Splitting Methods ................................ 95
3.2.5 Other Integration Methods ........................ 96
Operator Integration Factors....................... 96
Characteristics Methods ........................... 97
3.3 Homogeneous Flows .................................... 98
3.3.1 Fourier Galerkin Approximation
for Isotropic Turbulence........................... 98
3.3.2 De-aliasing Using Transform Methods............... 99
3.3.3 Pseudospectraland Collocation Methods ............ 103
3.3.4 Rogallo Transformation for Homogeneous Turbulence . 106
3.3.5 Large-Eddy Simulation of Isotropic Turbulence....... 108
3.3.6 The Taylor–GreenVortex Example:
Stability, Accuracy and Aliasing.................... 110
3.4 Flows with One Inhomogeneous Direction ................. 121
3.4.1 Coupled Methods ................................ 123
Kleiser–Schumann Algorithm ...................... 124
Normal Velocity–Normal Vorticity Algorithms........ 127
3.4.2 Galerkin Methods Using Divergence-Free Bases ...... 131
3.4.3 Splitting Methods ................................ 133
Chebyshev Staggered Grid ......................... 133
Zang–Hussaini Algorithm.......................... 135
3.4.4 Other Confined Flows............................. 138
Contents XI
3.4.5 Unbounded Flows ................................ 140
Flat-Plate Boundary-Layer Flows................... 140
Free-Shear-Layer Flows ........................... 142
3.4.6 A Numerical Example: Accuracy ................... 144
3.5 Flows with Multiple Inhomogeneous Directions............. 147
3.5.1 The Choice of Spatial Discretization in a Cavity...... 149
3.5.2 The Choice of Spatial Discretization
on a Reference Domain............................ 157
3.6 Outflow Boundary Conditions............................ 159
3.6.1 Fringe Regions ................................... 159
3.6.2 Buffer Domains .................................. 161
3.7 Analysis of Spectral Methods for Incompressible Flows ...... 162
3.7.1 Compatibility Conditions
Between Velocity and Pressure ..................... 165
3.7.2 Direct Discretization of the Continuity Equation:
The Inf-sup Condition ............................ 168
General Theory .................................. 169
3.7.3 Specific Applications.............................. 172
Numerical Results ................................ 177
Extensions....................................... 178
3.7.4 The Inf-sup Condition and the Pressure Operator .... 179
3.7.5 Discretizations of the Continuity Equation
by an Influence-Matrix Technique:
The Kleiser–SchumannMethod .................... 183
4. Single-Domain Methods for Compressible Flows .......... 187
4.1 Introduction ........................................... 187
4.2 Boundary Treatment for Hyperbolic Systems............... 187
4.2.1 Characteristic Compatibility Conditions............. 188
An Example of Unstable Treatment ................. 188
The Characteristic Compatibility Method (CCM) ..... 189
CCM for a General 1D System..................... 192
CCM for the Collocation Method ................... 193
CCM for a General Multidimensional System ........ 195
References and Outlook............................ 196
4.2.2 Boundary Treatment for Linear Systems
in Weak Formulations............................. 197
4.2.3 Spectral Accuracy and Conservation ................ 199
4.2.4 Analysis of Spectral Methods
for Symmetric Hyperbolic Systems ................. 200
4.3 Boundary Treatment for the Euler Equations .............. 203
4.4 High-Frequency Control ................................. 208
4.5 Homogeneous Turbulence................................ 211
4.5.1 Algorithmic Considerations ........................ 211
4.5.2 Representative Applications ....................... 214
