Innovative Algorithms and Analysis PDF
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Springer INdAM Series 16 Laurent Gosse Roberto Natalini Editors Innovative Algorithms and Analysis Springer INdAM Series Volume 16 Editor-in-Chief G.Patrizio SeriesEditors C.Canuto G.Coletti G.Gentili A.Malchiodi P.Marcellini E.Mezzetti G.Moscariello T.Ruggeri Moreinformationaboutthisseriesathttp://www.springer.com/series/10283 Laurent Gosse • Roberto Natalini Editors Innovative Algorithms and Analysis 123 Editors LaurentGosse RobertoNatalini IstitutoperleApplicazionidelCalcolo IstitutoperleApplicazionidelCalcolo CNR CNR Rome,Italy Rome,Italy ISSN2281-518X ISSN2281-5198 (electronic) SpringerINdAMSeries ISBN978-3-319-49261-2 ISBN978-3-319-49262-9 (eBook) DOI10.1007/978-3-319-49262-9 LibraryofCongressControlNumber:2017934879 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Foreword This book attempts to provide a fresh perspective on the well-posedness of hyperbolic models by highlighting some open questions and how some of them appearoraretreatedinsimilarinstances.Theseissueswerediscussedataworkshop entitledInnovativeAlgorithmsandAnalysisheldatINdAM(Rome)between17th and20thMay2016andorganizedbyLaurentGosseandRobertoNatalini.Thehope oftheorganizerswasthatthediversityoftheinvitedspeakerswouldhelptoidentify common themes in the study of nonclassical solutions to degenerate hyperbolic problems,balancelawswithnonlocalsourceterms,andweakadmissiblesolutions toconservationlawsinmultiplespacedimensions. The theory of hyperbolic conservation laws has matured significantly since the groundbreaking work of Bressan and Bianchini on the vanishing viscosity method;the stability analysisofBressan, Liu,and Yang;andthe study of viscous profilesbyZumbrun.Moreover,theworkofQiangandAmbrosiohasimprovedour understandingofthefunctionalframeworknecessaryfordiscussionofsingularities in transport problems. Numerics for conservation laws has also become quite sophisticated, although 1D formulations still serve as the basis for most multi-D discretizations.Manyofthebetterknownopenquestionsrelatetomultiscaleeffects in first-order hyperbolic problems, macroscopic limits, and Riemann problems in 2D. Nevertheless, a few problems in these subjects still persist in these subjects, particularly in 2D, such as the role of diffusion and entropy and the nature of singularities. In one space dimension, the theory of conservation laws is well understood, yet there remain a few unanswered issues. For example, in chapter “Analysis and Simulation of Nonlinear and Nonlocal Transport Equations”, Lagoutière and Vauchelet discuss measure-valued solutions appearing in nonlocal balance laws modeling aggregation, from both the analytical and the numerical perspective. MatthesandSöllnerstudydrift-diffusionwithnonlocalaggregationeffectsusingthe globalWassersteingradientflow,whichagainleadstosolutionsoutsidetheclassical framework.Karlsen,Risebro,andStorrøstenpresentafewexamplesofdegenerate conservationlawsforwhichtheobservednumericalconvergencerateforhigh-order schemesisgreaterthanthebestestimatesgivenbyKružkov’sstabilitytheory.These v vi Foreword results indicate that despite the success of the established stability theories, there remainphenomenajustbeyondtheclassicalBVframework. Christoforou provides in chapter “On Hyperbolic Balance Laws and Appli- cations” a concise and modern summary of the theory of balance laws in 1D, emphasizing necessary well-posedness conditions for the source terms. The dis- cussionidentifiesthestrengthsandweaknessesofdifferenttechniquesandcouldbe usefulto graduatestudentsenteringthe field.Theseresultsare thenappliedto the Gauss-Codazzisystemforthemetricofisometricimmersionsof2DsurfacesinR3, particularlyformetricswithminimalregularity.Relativitytheoryprovidesanumber ofinterestingproblemsforbalancelaws,asisfurtherhighlightedbythecontribution ofBini,Damour,andGeralicoonperturbationsofblackholespacetimes. Established stability theories for conservation laws, although able to identify asymptoticprofiles,lackpreciseinformationconcerningprevioussystemstatesand proceedwithroughapproximationsofwaveinteractions.GosseandZuazuapresent in chapter “Filtered Gradient Algorithms for Inverse Design Problems of One- DimensionalBurgersEquation”anopenproblemfromaerospaceengineeringthat requirestheidentificationofallinitialprofilesleadingtoagivenasymptoticprofile, which is something akin to a multivalued adjoint problem. Deeper understanding ofsuchaproblemcouldhelpinthecharacterizationoflossofinformationthrough shocks. In two space dimensions and beyond, the numerics of conservation laws have largelybeendoneunderthe assumptionthathigh-orderschemesin 1D,combined withdimensionalflux-splitting,aresufficienttogeneratenumericalapproximations which converge to a class of unique solutions, something akin to the a poste- riori conclusion of the Lax-Richtmyer theorem. Unfortunately, the absence of a general well-posedness theory in multi-D does not warrant such a casual attitude toward numerical methods in 2D. In chapter “A Numerical Glimpse at Some Non-standard Solutions to Compressible Euler Equations”, Chiodaroli and Gosse attempt to numerically reproduce nonuniqueness in 2D solutions for isentropic flowsbyexploitingearlierexplicitconstructionsofnonuniquenessbyDeLellisand Szekelyhidi. As was already pointed out by Volker Elling, the nonuniqueness is causedbyunboundedvorticity,butevenwhenentropyadmissibilityissatisfied. Numerical MHD developers, who work almost exclusively in multi-D, have always been strong proponents for the use of multi-D solvers, that is, solvers without dimensionalsplitting. In chapter “A Well-Balanced Scheme for the Euler Equations with Gravitation”, Käppeli presents a well-balanced scheme that pre- servestheisothermalhydrostaticequilibrium.Hisworkrequiresthestandardlocal reconstructionproceduretobere-examinedinordertoensurethattheequilibrium ispreserved,inconjunctionwithcompatiblediscretizationsofthepotential.Gosse alsopresents,inchapter“ViscousEquationsTreatedwithL-SplinesandSteklov- PoincaréOperatorintwoDimensions”,are-interpretationofmanyclassicalsolvers usinglocalapproximationsthataresolutionstolocalanalogsoftheglobalproblem. Inbothcases,theauthorsfindfaultinreconstructionsbasedonunphysicalcriteria. As the authors of this book have made clear, some fundamental insights into hyperbolicproblemsare still lacking. It was the hope of the workshop organizers Foreword vii that by collecting these works into a monograph, this would encourage the community to re-examine some of these problems. It is this authors’ belief that thismonographwillgoalongwaytowardreachingthosegoals. Montréal,Canada MarcLaforest September2016 Preface Backin1978,nothingworked.Nowin 2013,almosteverythingworks,but nothingworksparticularlywell. ProfessorofAerospaceEngineering PHILIPL.ROE Historically, algorithm development may be seen as an iterative optimization process subject to three constraints, severe and perhaps contradictory with each other: (cid:129) Accuracy,thatis,themaximumdeviationbetweenthenumericalapproximation and the exact solution of the continuousmodel (assuming it is well posed, we shallcomebacktothatissuelater!)theuseriswillingtotolerate (cid:129) Stability, which usually refers to the ability of a constructive process not to diverge when the mesh size (or another “small parameter”) is arbitrarily decreased (cid:129) Efficiency,orsimplythecostofexecution,intermsofCPUflops A cornerstone result proved by Peter Lax, the so-called “Equivalence Theorem”, states that satisfying the first two criteria is a sufficientcondition for convergence ofthesequenceofapproximations.Thethirdcriterionhasameaningwhichvaries alotaccordingto theepoch:nowadays,“costly”referstotasks, thecomplexityof whichwouldhavebeenunthinkable1duringCourant’s,Richtmyer’s,orSamarskii’s time. A concrete manifestation of “cost constraints” was the introduction of so- calledvariationalfinite-differences,asabranchofRitz-typeapproximationmethods endowedwithasufficientlysmallstencil,soastokeeponallowingarathercheap inversionoftheresulting(sparse)massmatrix,evenathighordersofaccuracy. Despitecertainrefinements,likeuniform-typeestimatesoriginallydevelopedfor convection-dominated transport (nowadays rephrased as “asymptotic-preserving” 1Newareasofresearch,like“BigData,”arepartlyspawnedbycontemporarycomputingpower. ix x Preface properties), the notion of “stability” is rather easy to apprehend; it can, though, be an arduous task to pick the right norm in which uniform bounds are to be sought.However,whatreallyliesbehindtheterm“accuracy”canbemorepuzzling and context dependent. The more elementary notion of accuracy relies on both truncation errors and Taylor’s expansions, that is, on the density of polynomials in reasonable spaces of smooth, at least continuous, functions (Stone-Weierstrass theorem).However,whenitcomestohandlingdifferentialoperatorsendowedwith either weak solutions or advanced geometric features, such a notion has many shortcomings;researchershenceforthdevelopedalternative,moreelaboratenotions ofaccuracy,including: 1. Symplectic integrators, typically for Hamiltonian systems in which longtime preservationofcertainqualitativefeaturesofphase-spacetrajectoriesiscrucial 2. Mimetic finite differences for multidimensional diffusion equations, where, given a strongly nonuniform gridding of a complex geometry, one is led to definecompatible(possiblyhigh-order)discretizationsofubiquitousdifferential operators,likediv, r,andcurl,in sucha waythatdualitypairings(likeStokes formula),holdingatthecontinuouslevel,arepreservedatadiscreteone 3. Consistency with both the original equation and a supplementary differential operator, of frequent use in continuum mechanics, where physically relevant (weak)solutionsarebelievedtobeidentifiedthankstoan“entropyinequality” These simple examples(to which mighthave beenappended,e.g.,hypocoercivity estimates) show that, through the passing of time, researchers switched from a notionofaccuracy,asbeingmeasuredmostlythroughremainingtermsinducedby finite-differenceapproximationsofpartialderivatives,tothenotionof preservation of either qualitative or quantitative features displayed by the exact solution of the underlying problem, which is really morechallenging,in the sense thatit requiresathoroughknowledgeoftheexactsolution’speculiaritiesinordertowisely decidewhichonesaresosalientastobenecessaryinitsdiscretecounterpart. Accuracy ! ExactSolution’sFeaturesPreservation Clearly, in a context of, say, diffusion equations with rough, position-dependent coefficients, possibly admitting weak solutions endowed with sharp transition layers,usualtruncationestimatesgivelittle,ifany,usefulinformation;thereasonis thatonereallyneedserrorboundsdependingaslessaspossibleonthecoefficients’ smoothness.Hence,theneedofsubstitutingusualpolynomialbases(inbothfinite elements and finite differences) with more suitable, problem-fit functions was alreadywellidentifiedbyTrefftzinhisseminal1926paper:“Insteadofusingtrial functionsthatsatisfy theboundaryconditionsbutviolatethedifferentialequation, we approximate the solution by selecting functions that violate the boundary conditions but satisfy the differential equation. We can view this procedure as a generalizationofseriesexpansionsusingparticularsolutionsandthenewapproach infactcontainstheconventionalseriesexpansionsasaspecialcase.”Suchanidea
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