Courant–Friedrichs–Lewy Condition - Wikipedia, The Free Encyclopedia
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Courant–Friedrichs–Lewy condition From Wikipedia, the free encyclopedia In mathematics, the Courant–Friedrichs–Lewy (CFL) condition is a necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically by the method of finite differences. [1] It arises in the numerical analysis of explicit time integration schemes, when these are used for the numerical solution. As a consequence, the time step must be less than a certain time in many explicit timemarching computer simulations, otherwise the simulation will produce incorrect results. The condition is named after Richard Courant, Kurt Friedrichs, and Hans Lewy who described it in their 1928 paper. [2] Contents 1 Heuristic description 2 The CFL condition 2.1 The onedimensional case 2.2 The two and general ndimensional case 3 Implications of the CFL condition 3.1 The CFL condition is only a necessary one 3.2 The CFL condition can be a very strong requirement 4 Notes 5 References 6 External links Heuristic description The principle behind the condition is that, for example, if a wave is moving across a discrete spatial grid and we want to compute its amplitude at discrete time steps of equal duration, [3] then this duration must be less than the time for the wave to travel to adjacent grid points. As a corollary, when the grid point separation is reduced, the upper limit for the time step also decreases. In essence, the numerical domain of dependence of any point in space and time (as determined by initial conditions and the parameters of the approximation scheme) must include the analytical domain of dependence (wherein the initial conditions have an effect on the exact value of the solution at that point) in order to assure that the scheme can access the information required to form the solution. The CFL condition In order to make a reasonably formally precise statement of the condition, it is necessary to define the following quantities Spatial coordinate: it is one of the coordinates of the physical space in which the problem is posed. Spatial dimension of the problem: it is the number of spatial dimensions i.e. the number of spatial coordinates of the physical space where the problem is posed. Typical values are , and . Time: it is the coordinate, acting as a parameter, which describes the evolution of the system, distinct from the spatial coordinates.
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Transcript of Courant–Friedrichs–Lewy Condition - Wikipedia, The Free Encyclopedia
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24/05/2015 CourantFriedrichsLewy condition - Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/Courant%E2%80%93Friedrichs%E2%80%93Lewy_condition 1/4
CourantFriedrichsLewyconditionFromWikipedia,thefreeencyclopedia
Inmathematics,theCourantFriedrichsLewy(CFL)conditionisanecessaryconditionforconvergencewhilesolvingcertainpartialdifferentialequations(usuallyhyperbolicPDEs)numericallybythemethodoffinitedifferences.[1]Itarisesinthenumericalanalysisofexplicittimeintegrationschemes,whentheseareusedforthenumericalsolution.Asaconsequence,thetimestepmustbelessthanacertaintimeinmanyexplicittimemarchingcomputersimulations,otherwisethesimulationwillproduceincorrectresults.TheconditionisnamedafterRichardCourant,KurtFriedrichs,andHansLewywhodescribeditintheir1928paper.[2]
Contents
1Heuristicdescription2TheCFLcondition
2.1Theonedimensionalcase2.2Thetwoandgeneralndimensionalcase
3ImplicationsoftheCFLcondition3.1TheCFLconditionisonlyanecessaryone3.2TheCFLconditioncanbeaverystrongrequirement
4Notes5References6Externallinks
Heuristicdescription
Theprinciplebehindtheconditionisthat,forexample,ifawaveismovingacrossadiscretespatialgridandwewanttocomputeitsamplitudeatdiscretetimestepsofequalduration,[3]thenthisdurationmustbelessthanthetimeforthewavetotraveltoadjacentgridpoints.Asacorollary,whenthegridpointseparationisreduced,theupperlimitforthetimestepalsodecreases.Inessence,thenumericaldomainofdependenceofanypointinspaceandtime(asdeterminedbyinitialconditionsandtheparametersoftheapproximationscheme)mustincludetheanalyticaldomainofdependence(whereintheinitialconditionshaveaneffectontheexactvalueofthesolutionatthatpoint)inordertoassurethattheschemecanaccesstheinformationrequiredtoformthesolution.
TheCFLcondition
Inordertomakeareasonablyformallyprecisestatementofthecondition,itisnecessarytodefinethefollowingquantities
Spatialcoordinate:itisoneofthecoordinatesofthephysicalspaceinwhichtheproblemisposed.Spatialdimensionoftheproblem:itisthenumber ofspatialdimensionsi.e.thenumberofspatialcoordinatesofthephysicalspacewheretheproblemisposed.Typicalvaluesare ,
and .Time:itisthecoordinate,actingasaparameter,whichdescribestheevolutionofthesystem,distinctfromthespatialcoordinates.
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24/05/2015 CourantFriedrichsLewy condition - Wikipedia, the free encyclopedia
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Thespatialcoordinatesandthetimearesupposedtobediscretevaluedindependentvariables,whichareplacedatregulardistancescalledtheintervallength[4]andthetimestep,respectively.Usingthesenames,theCFLconditionrelatesthelengthofthetimesteptoafunctionoftheintervallengthsofeachspatialcoordinateandofthemaximumspeedwithwhichinformationcantravelinthephysicalspace.
Operatively,theCFLconditioniscommonlyprescribedforthosetermsofthefinitedifferenceapproximationofgeneralpartialdifferentialequationswhichmodeltheadvectionphenomenon.[5]
Theonedimensionalcase
Foronedimensionalcase,theCFLhasthefollowingform:
wherethedimensionlessnumberiscalledtheCourantnumber,
isthemagnitudeofthevelocity(whosedimensionislength/time)isthetimestep(whosedimensionistime)isthelengthinterval(whosedimensionislength).
Thevalueof changeswiththemethodusedtosolvethediscretisedequation,especiallydependingonwhetherthemethodisexplicitorimplicit.Ifanexplicit(timemarching)solverisusedthentypically .Implicit(matrix)solversareusuallylesssensitivetonumericalinstabilityandsolargervaluesof maybetolerated.
Thetwoandgeneralndimensionalcase
Inthetwodimensionalcase,theCFLconditionbecomes
withobviousmeaningofthesymbolsinvolved.Byanalogywiththetwodimensionalcase,thegeneralCFLconditionforthe dimensionalcaseisthefollowingone:
Theintervallengthisnotrequiredtobethesameforeachspatialvariable .This"degreeoffreedom"canbeusedinordertosomewhatoptimizethevalueofthetimestepforaparticularproblem,byvaryingthevaluesofthedifferentintervalinordertokeepitnottoosmall.
ImplicationsoftheCFLcondition
TheCFLconditionisonlyanecessaryone
TheCFLconditionisanecessarycondition,butmaynotbesufficientfortheconvergenceofthefinitedifferenceapproximationofagivennumericalproblem.Thus,inordertoestablishtheconvergenceofthefinitedifferenceapproximation,itisnecessarytouseothermethods,whichinturncouldimply
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24/05/2015 CourantFriedrichsLewy condition - Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/Courant%E2%80%93Friedrichs%E2%80%93Lewy_condition 3/4
furtherlimitationsonthelengthofthetimestepand/orthelengthsofthespatialintervals.
TheCFLconditioncanbeaverystrongrequirement
TheCFLconditioncanbeaverylimitingconstraintonthetimestep :forexample,inthefinitedifferenceapproximationofcertainfourthordernonlinearpartialdifferentialequations,itcanhavethefollowingform:
meaningthatadecreaseinthelengthinterval requiresafourthorderdecreaseinthetimestepfortheconditiontobefulfilled.Therefore,whensolvingparticularlystiffproblems,effortsareoftenmadetoavoidtheCFLcondition,forexamplebyusingimplicitmethods.
Notes
References
Courant,R.Friedrichs,K.Lewy,H.(1928),"berdiepartiellenDifferenzengleichungendermathematischenPhysik"(http://resolver.sub.unigoettingen.de/purl?GDZPPN002272636),MathematischeAnnalen(inGerman)100(1):3274,Bibcode:1928MatAn.100...32C(http://adsabs.harvard.edu/abs/1928MatAn.100...32C),doi:10.1007/BF01448839(https://dx.doi.org/10.1007%2FBF01448839),JFM54.0486.01(https://zbmath.org/?format=complete&q=an:54.0486.01),MR1512478(https://www.ams.org/mathscinetgetitem?mr=1512478).Courant,R.Friedrichs,K.Lewy,H.(September1956)[1928],Onthepartialdifferenceequationsofmathematicalphysics(http://www.archive.org/stream/onpartialdiffere00cour#page/n0/mode/2up),AECResearchandDevelopmentReport,NYO7689,NewYork:AECComputingandAppliedMathematicsCentreCourantInstituteofMathematicalSciences,pp.V+76,archivedfromtheoriginal(http://www.archive.org/details/onpartialdiffere00cour)onOctober23,2008.:translatedfromtheGermanbyPhyllisFox.ThisisanearlierversionofthepaperCourant,Friedrichs&Lewy1967,circulatedasaresearchreport.Courant,R.Friedrichs,K.Lewy,H.(March1967)[1928],"Onthepartialdifferenceequationsofmathematicalphysics"(http://domino.research.ibm.com/tchjr/journalindex.nsf/a3807c5b4823c53f85256561006324be/769774a3c9f3685f85256bfa00683f8a!OpenDocument),IBMJournalofResearchandDevelopment(http://researchweb.watson.ibm.com/journal/rdindex.html)11(2):215234,MR0213764(https://www.ams.org/mathscinetgetitem?mr=0213764),Zbl0145.40402(https://zbmath.org/?format=complete&q=an:0145.40402).Afreelydownlodablecopycanbefoundhere(http://www.stanford.edu/class/cme324/classics/courantfriedrichslewy.pdf).
1. Ingeneral,itisnotasufficientconditionalso,itcanbeademandingconditionforsomeproblems.Seethe"ImplicationsoftheCFLcondition"sectionofthisarticleforabriefsurveyoftheseissues.
2. SeereferenceCourant,Friedrichs&Lewy1928.ThereexistsalsoanEnglishtranslationofthe1928Germanoriginal:seereferencesCourant,Friedrichs&Lewy1956andCourant,Friedrichs&Lewy1967.
3. Thissituationcommonlyoccurswhenahyperbolicpartialdifferentialoperatorhasbeenapproximatedbyafinitedifferenceequation,whichisthensolvedbynumericallinearalgebramethods.
4. Thisquantityisnotnecessarilythesameforeachspatialvariable,asitisshowninthe"Thetwoandgeneralndimensionalcase"sectionofthisentry:itcanbechoseninordertosomewhatrelaxthecondition.
5. Precisely,thisisthehyperbolicpartofthePDEunderanalysis.
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24/05/2015 CourantFriedrichsLewy condition - Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/Courant%E2%80%93Friedrichs%E2%80%93Lewy_condition 4/4
Externallinks
Bakhvalov,N.S.(2001),"CourantFriedrichsLewycondition"(http://www.encyclopediaofmath.org/index.php?title=C/c026760),inHazewinkel,Michiel,EncyclopediaofMathematics,Springer,ISBN9781556080104Weisstein,EricW.,"CourantFriedrichsLewyCondition"(http://mathworld.wolfram.com/CourantFriedrichsLewyCondition.html),MathWorld.
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Categories: Numericaldifferentialequations Computationalfluiddynamics
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