Table Of ContentSpringer 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
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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
