Table Of ContentHandbook of Numerical Analysis
Series Editors
Qiang Du
Columbia University, New York, United States of America
Roland Glowinski
University of Houston, Texas, United States of America
€
Michael Hintermuller
Humboldt University of Berlin, Germany
€
Endre Suli
University of Oxford, United Kingdom
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Contributors
NumbersinParenthesesindicatethepagesonwhichtheauthor’scontributionsbegin.
A.Abdulle(545),ANMC,SectiondeMath(cid:1)ematiques,E(cid:1)colepolytechniquef(cid:1)ed(cid:1)eralede
Lausanne, Lausanne, Switzerland
R. Abgrall (351, 507), Institute of Mathematics, University of Zu€rich, Zu€rich,
Switzerland
S. C(cid:3)ani(cid:1)c(435),University ofHouston, Houston,TX,United States
M.Berger(1),CourantInstitute,NewYorkUniversity,NewYork,NY,UnitedStates
M.J. Castro (131),Universidad deMa´laga, Ma´laga, Spain
A.Chertock (177),North Carolina StateUniversity, Raleigh, NC,UnitedStates
M.L.DelleMonache(435),RutgersUniversity—Camden,Camden,NJ,UnitedStates
I.M. Gamba(403),The University ofTexas atAustin,Austin, TX,United States
H. Guillard(203),Universit(cid:1)eC^oted’Azur, Inria,CNRS, LJAD,France
P. Henning(545),KTHRoyal Instituteof Technology, Stockholm,Sweden
P. Houston (233), School of Mathematical Sciences, University of Nottingham,
Nottingham, United Kingdom
J. Hu(103),Purdue University, West Lafayette, IN, UnitedStates
A.Jameson (303),Stanford University, Stanford,CA, UnitedStates
S. Jin(103),University ofWisconsin-Madison, Madison, WI, UnitedStates
C. Klingenberg (465), Institut fu€r Mathematik, Universit€at Wu€rzburg, Wu€rzburg,
Germany
Q. Li (103), Universityof Wisconsin-Madison, Madison,WI, UnitedStates
A.Loseille (263),INRIA Saclay-Ile de France,France
S. Mishra (479, 507), Seminar for Applied Mathematics, ETH Zu€rich, Zu€rich,
Switzerland
T. Morales de Luna(131),Universidad de Co´rdoba,Co´rdoba,Spain
C.-D. Munz (385), Institute for Aerodynamics and Gas Dynamics, University of
Stuttgart, Stuttgart,Germany
B. Nkonga(203),Universit(cid:1)eC^oted’Azur,Inria,CNRS, LJAD, France
C.Pare(cid:1)s(131),Universidad de Ma´laga,Ma´laga,Spain
B. Piccoli (435),RutgersUniversity—Camden, Camden, NJ, UnitedStates
xv
xvi Contributors
J.-M.Qiu(435),Universityof Houston,Houston, TX,UnitedStates
P.Roe(53),Universityof Michigan, AnnArbor, MI,UnitedStates
C.-W.Shu(23),Brown University, Providence,RI, UnitedStates
E. Sonnendru€cker (385), Max-Planck Institute for Plasma Physics; Mathematics
Center,TU Munich,Garching, Germany
J.Tambacˇa (435),Universityof Zagreb, Zagreb, Croatia
S.Tan (23),Brown University, Providence,RI, UnitedStates
F.D.Witherden (303),Stanford University, Stanford, CA,UnitedStates
Y.Xing (361), UniversityofCalifornia Riverside, Riverside, CA, UnitedStates
Z.Xu (81),MichiganTechnological University, Houghton, MI,UnitedStates
X.Zhang(81),Purdue University, West Lafayette, IN, UnitedStates
D.W.Zingg(303),Universityof TorontoInstitutefor Aerospace Studies, Toronto,
ON,Canada
Editors’ Introduction
R. Abgrall* and C.-W. Shu†
*Institutfu€rMathematik,Universita€tZu€rich,Zu€rich,Switzerland
†BrownUniversity,Providence,RI,UnitedStates
Thesetwovolumesrepresent thevolumes17 and18ofHandbook ofNumer-
ical Analysis. It is entirely devoted to the numerical methods designed for
approximatingthesolutionofhyperbolicequations,orofequationsthatwrite
as a sum of operators where the most important, in terms of the behaviour of
the solution, is the hyperbolic one. An example is the Navier–Stokes equa-
tions with high Reynolds number where the solution behaviour is essentially
dictated by the hyperbolic operator (here the Euler system), except in bound-
ary layers because of the boundary conditions.
Hyperbolic partial differential equations appear often in applications. The
most important application, already mentioned, is fluid dynamics, including
specific flows such as multiphase flows, magnetohydrodynamics, water
waves, etc. Other application areas include Maxwell equations, kinetic equa-
tions, traffic flow models and networks, etc. The solutions of hyperbolic
partial differential equations often involve discontinuities, making mathemat-
ical analysis and numerical simulations difficult. In the past few decades
therehasbeenalargeamountofliteratureinthedesign,analysisandapplica-
tion of various numerical algorithms for solving hyperbolic equations. The
currentvolumesattempttohaveexpertsindifferenttypesofalgorithmswrite
concise summaries so that the readers can find a variety of algorithms under
different situations and become familiar with their relative advantages and
limitations.
This is a formidable task. We had to make choices because the field has
grown tremendously since the early ages dating back to von Neumann in
the United States and researchers from the former Soviet Union such as
Rusanov and Godunov. This field has grown up for various reasons. The
demand on diverse high tech areas ranging from airplanes and rockets, to
the nuclear and car industries as well as more recently the green industry, to
namejustafew,necessitates tomasterbetterandbettertoolstoimproveper-
formance. If it was possible in the early ages to rely on analytical solutions
and experimental facilities only, this is no longer the case because of various
constraints: economical, technological (weight, etc.), energy consumption,
etc. This evolution has needed improved algorithms, i.e., more and more
xvii
xviii Editors’Introduction
accurate as well as more and more robust ones. Hence the research on algo-
rithm has grown up and then exploded since the early 1970s.
In parallel, and also triggered by the same needs, computers have been
more and more powerful from scalar, to vectors, then parallel and now mas-
sivelyparallelandhybridarchitectures.Thisevolutionoftechnologyhasalso
had a strong impact on the algorithms development.
Becauseofitssuccess,itisnowpossibletocomputemoreandmorecom-
plicated problems, both in terms of geometry and physics.
Thereisstillalottodotoimproveandunderstandthenumericalmethods
designed for hyperbolic problems. The aim of these two volumes is to give a
picture of the current state of the art.
Inorder tointroducethe subject, we have asked Professor Dafermos from
BrownUniversitytoprovideashortsummaryonthetheoryofhyperbolicequa-
tions.Then,ifonelooksatthetableofcontent,onewouldrealizethatwehave
triedtocovernotonlytheclassicaltopics,suchasthefinitevolumemethodand
theRiemannsolversthatarethebuildingblocksofmanyofthealgorithms,but
alsolessstandardmethods.Examplesincludealgorithmsforcomputingsharp
transitionpropagatedbylinearlydegeneratewaves.Otherexamplesaregiven
bytheENO/WENOfamily.Inthatcasewehavetriedtogoovertheclassical
description,bygivingsomeanalysisofthemethods.Otherhigh-ordermethods
are also considered such as the discontinuous Galerkin (DG) ones, the more
recent hybrid DG schemes, high-order finite element methods, front-tracking
methods, methods for Lagrangian hydrodynamics, entropy stable schemes,
etc.Timediscretisationisalsoconsidered,aswellasmorespecializedproblems
like the simulation of flows with low Mach numbers, level set techniques,
numericalmethodsforHamilton–Jacobiequations,etc.
Unfortunately,itisnotpossible,evenintwoquitethickvolumes,topro-
vide an exhaustive coverage of the state of the art. Even though the table of
content seems to be exhaustive, many topics are still missing. For example,
we have chosen to be quite restrictive on the subject of time stepping: there
is no coverage on ADER and IMEX methods. The handling of problems
with source terms is touched by chapters 5 and 6 (well-balanced schemes
and asymptotic-preserving schemes), but there is no direct coverage on stiff
sourceterms.IfwehaveachapteronmethodsforCartesianmeshes,thereis
nodirectcoverageontheapplicationofimmersedboundarymethods.Simi-
larly we have chosen to consider the problem of meshing in a specific way;
there is no direct coverage on adaptive mesh refinement (AMR). The prob-
lem on boundary conditions is considered in chapter 2 this volume (volume
18) and chapter 19 previous volume (volume 17) of the handbook (SAT-
SPB schemes and inverse Lax–Wendroff procedure), but much more could
have been said. It was simply impossible to cover the whole field, and we
apologize for this.
To end this introduction, we would like to thank all the contributors to
these volumes, as well as the referees. Both have been extremely efficient.
Editors’Introduction xix
ACKNOWLEDGEMENTS
R.A. has been supported in part by SNF grant # 200021_153604. C.-W.S. has been sup-
portedinpartbyNSFgrantDMS-1418750.
Chapter 1
Cut Cells: Meshes and Solvers
M. Berger
CourantInstitute,NewYorkUniversity,NewYork,NY,UnitedStates
Chapter Outline
1 Introduction 1 5.1 Steady-StateSolution
2 BriefEarlyHistory 3 Techniques 12
3 MeshGeneration 5 5.2 ExplicitTime-dependent
4 DataStructuresand SolutionTechniques 13
ImplementationIssues 8 5.3 ViscousFlows 17
5 FiniteVolumeMethodsfor 6 Conclusions 18
CutCells 10 Acknowledgements 18
References 18
ABSTRACT
EmbeddedboundaryCartesiangridshavegreatlysimplifiedthegridgenerationprocess
forcomplicatedengineeringgeometries.Thisapproachtomeshgenerationusesaback-
ground regular mesh andtakes special care of cutcells where the geometry intersects
thegrid.Gridgenerationisfastandrobust.Thischaptergivesanoverviewofcut-cell
grids. We discuss mesh generation and data structures used for implementation. We
describesomestableandaccuratediscretizationsforthesmallirregularcellsthatoccur
attheboundary.Muchofthistechnologyismatureandformsthebasisforwidelyused
CFDpackages.Somepossibledirectionsforfutureresearcharepresented.
Keywords:Cut cells,Embeddedboundarymeshes,Smallcellproblem
AMSClassification Codes:65M08,65M12,65M50,76M12
1 INTRODUCTION
ThischapterdescribesthedevelopmentandcurrentstatusofCartesiancut-cell
meshgenerators andsolvers. Mesh generationcan bethe mostexpensive and
time-consumingpartoftheflowsolutionprocess,particularlyforcomplexengi-
neeringgeometriesfromCADsystems.Cartesianmeshmethodssimplifyand
automatethegridgenerationprocessbyusingaregularCartesianmeshthatis
notalignedwiththegeometry.Thisreducestheprocessofmeshgenerationto
HandbookofNumericalAnalysis,Vol.18.http://dx.doi.org/10.1016/bs.hna.2016.10.008
©2017ElsevierB.V.Allrightsreserved. 1
2 HandbookofNumericalAnalysis
computing intersections of the geometry and the Cartesian mesh on the
lower-dimensionalsurfacewherethegeometryintersectstheCartesianmesh.
There are many advantages to embedded boundary meshes, and a few
notable difficulties. The mesh generation process is fast, easy to automate,
and can handle configurations of arbitrary complexity. The meshes are also
verycomputationallyefficient:typically90%ofthemeshisregularCartesian
cells, with no metric terms to store or use. Discretizations on Cartesian
meshes are also more accurate than their unstructured counterparts
(Aftosmis and Berger, 2002). The difficulties lie with the cut cells. Cut cells
are much more irregular than unstructured meshes. There is no grid smooth-
ness at cut cells, and so there error depends on the mesh in a complex way.
This makes it difficult to estimate truncation error using Richardson-type
methods. The use of explicit methods to update the solution on cut cells is
problematic due to their arbitrarily small size. Perhaps the biggest problem
is the lack of anisotropic refinement, or boundary layer zoning, for
noncoordinate-aligned bodies. This makes embedded boundary grids hard to
apply for computing high Reynolds number flow, a current research topic.
Allofthesepointswillbediscussedinmoredetailinthesectionsthatfollow.
Cartesianmeshmethodsdifferintheirmeshassumptions,geometryrepre-
sentation, data structures, and solution techniques, including what equations
are solved and how. The goal of this chapter of the handbook is to survey
the range of these approaches. We will not be able to go into much detail,
but hope to provide enough context and references to bring a reader new to
Cartesian mesh methods to the frontier today.
The terminology used to describe embedded boundary Cartesian meshes
varies widely, particularly when describing cut cell vs immersed boundary
meshes.Inourdefinition,acut-cellmeshcomputesdetailedintersectionswith
the geometry, and actually removes the portion of the mesh that lies outside
theflowfield.Thisleavesaborderofhighlyirregularcellsontheremainingpart
oftheCartesianmesh.Theflowcomputationisdoneonlyontheseremaining
flow cells, and the solid wall boundary conditions are applied only to the cut
cellsthatarestillpartofthemesh.Bycontrast,theoriginalimmersedboundary
method was proposed by Peskin in 1972 (Peskin, 1972, 1977) for biological
fluidflows.Heusedaseriesofnodesandfibrestorepresenttheheart,superim-
posedonaCartesianmesh.Thefibresrepresentedatwo-dimensionalheart,and
bothsidesofthestructurewerefluid.Thefibrousinterfaceinteractedwiththe
Cartesianmeshbyapplyingforces,discretizedoverasetofpointsontheregular
meshintheneighbourhoodsurroundingbothsidesoftheinterface.
Inmorerecentimmersedboundaryresearch,theboundaryhasbeenrepre-
sented using level sets (a very nice survey article is Mittal and Iaccarino
(2005)). For greater accuracy in interface calculations, more detailed mesh
intersection quantities have been computed. It is no longer always assumed
that both sides of an interface are part of a calculation. Similarly, some
cut-cell researchers have used level sets to represent the geometry, to make
it easier, for example, to compute curvature. The term embedded boundary
CutCells:MeshesandSolvers Chapter 1 3
FIG.1 Exampleofanadaptivecut-cellmeshforthelaunchabortvehicle,acomplicatedconfig-
urationwithgridfins,abortmotors,andmultiplelengthscales.ComputationbyM.Aftosmis.
mesh is now in common usage, and covers all these types of nonbody-fitted
meshes. Often, the term Cartesian mesh is used by itself, and the type is not
even specified. We will still mostly use the term cut-cell mesh to refer to a
Cartesian mesh with the geometry cut out from the mesh, as shown in
Fig. 1, where no computation is performed on the solid side. There are many
issuesincommon,however,andthedemarcationbetweenthetwoapproaches
is disappearing.
Thischapterisorganizedasfollows.Section2givesaverybriefhistoryof
thedevelopmentofcutcells,withapologiesinadvanceforthosethatareacci-
dentallyoverlooked.Sections3and4discussissuesrelatedtomeshgeneration
anddatastructures,tworelatedconcerns.Section5discussesflowsolversfor
cut-cellmeshes.Manyinterestingmathematicalquestionsarisebecauseofthe
irregularityofcutcells,andtheirpossiblytinycellvolumes.Someconclusions
andopenproblemsareinSection6.
2 BRIEF EARLY HISTORY
TheearliestworkusingCartesianmeshesintersectedcoordinatelineswiththe
geometry to build the mesh, and solved using finite difference methods mod-
ified by Shortley–Weller formulas to account for the uneven mesh spacing at
thegeometryboundary.Thisearliestworksolvedellipticequations,sostabil-
ity of these modified schemes was not an issue, although conditioning could
be affected. (In current work solving hyperbolic equations, however, it is a
big issue, discussed in Section 5.2.) In other very early work, Cartesian
meshes used a piecewise constant discretization of the geometry, resulting
in so-called staircase meshes. A cell was declared to be part of the computa-
tional mesh or conversely inside the geometry, depending on a simple test.
For example, the test could be whether the cell centroid, or alternatively at
