Handbook of Numerical Methods for Hyperbolic Problems Applied and Modern Issues PDF
Preview Handbook of Numerical Methods for Hyperbolic Problems Applied and Modern Issues
Handbook 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 North-HollandisanimprintofElsevier Radarweg29,POBox211,1000AEAmsterdam, TheNetherlands TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom Copyright©2017Elsevier B.V.Allrightsreserved. Nopart ofthispublicationmaybereproducedortransmittedinanyformorbyany means,electronic ormechanical, includingphotocopying,recording, orany informationstorageandretrieval system, withoutpermissioninwritingfromthe publisher.Detailsonhowtoseekpermission,furtherinformationaboutthePublisher’s permissionspoliciesandourarrangementswithorganizationssuchastheCopyright Clearance CenterandtheCopyrightLicensingAgency, canbefoundatourwebsite: www.elsevier.com/permissions. 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ISBN:978-0-444-63910-3 ISSN:1570-8659 ForinformationonallNorth-Holland publications visitourwebsiteat https://www.elsevier.com/ Publisher:Zoe Kruze AcquisitionEditor:KirstenShankland EditorialProjectManager:HannahColford ProductionProjectManager:StalinViswanathan CoverDesigner: VictoriaPearson TypesetbySPi Global,India 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
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