courant-friedrichs-lewy

R.Courant*K.Friedrichs”H.LewytOnthePartialDifferenceEquationsofMathematicalPhysicsEditor’snote:Thispaper,whichoriginallyappearedinMathematischeAnnalen100,32-74(1928),isrepublishedbypermissionoftheauthors.WearealsogratefultotheAtomicEnergyCommissionforpermissiontorepublishthistranslation,whichhadappearedasAECReportNYO-7689,andtoPhyllisFox,thetranslator,whodidtheworkattheAECComputingFacilityatNewYorkUniversityunderAECContractNo.AT(30-1)-1480.ProfessorEugeneIsaacsonhadmadesuggestionsonthistranslation.IntroductionProblemsinvolvingtheclassicallinearpartialdifferentialequationsofmathematicalphysicscanbereducedtoalgebraiconesofaverymuchsimplerstructurebyreplac-ingthedifferentialsbydifferencequotientsonsome(sayrectilinear)mesh.Thispaperwillundertakeanelementarydiscussionofthesealgebraicproblems,inparticularofthebehaviorofthesolutionasthemeshwidthtendstozero.Forpresentpurposeswelimitourselvesmainlytosimplebuttypicalcases,andtreattheminsuchawaythattheapplicabilityofthemethodtomoregeneraldifferenceequationsandtothosewitharbitrarilymanyindependentvariablesismadeclear.Correspondingtothecorrectlyposedproblemsforpartialdifferentialequationswewilltreatboundaryvalueandeigenvalueproblemsforellipticdifferenceequations,andinitialvalueproblemsforthehyperbolicorparaboliccases.Wewillshowbytypicalexamplesth,atthepassagetothelimitisindeedpossible,i.e.,thatthesolutionofthedifferenceequationconvergestothesolutionofthecorrespondingdifferentialequation;infactwewillfindthatforellipticequationsingeneraladifferencequotientofarbitrarilyhighordertendstothecorrespondingderiv-ative.Nowheredoweassumetheexistenceofthesolutiontothedifferentialequationproblem-onthecontrary,weobtainasimpleexistenceproofbyusingthelimitingprocess.’Forthecaseofellipticequationsconvergenceis1Ourmethodofproofmaybeextendedwithoutdifficultytocoverbound-aryvalueandeigenvalueproblemsforarbitrarylinearellipticdifferentialequationsandinitialvalueproblemsforarbitrarylinearhyperbolicdifferentialequations.obtainedindependentlyofthechoiceofmesh,butwewillfindthatforthecaseoftheinitialvalueproblemforhyperbolicequations,convergenceisobtainedonlyiftheratioofthemeshwidthsindifferentdirectionssatis-fiescertaininequalitieswhichinturndependontheposi-tionofthecharacteristicsrelativetothemesh.Wetakeasatypicalcasetheboundaryvalueproblemofpotentialtheory.Itssolutionanditsrelationtothesolutionofthecorrespondingdifferenceequationhasbeenextensivelytreatedduringthepastfewyears.’How-everincontrasttothepresentpaper,thepreviousworkhasinvolvedtheuseofquitespecialcharacteristicsofthepotentialequationsothattheapplicabilityofthemethodusedtheretootherproblemshasnotbeenimmediatelyevident.Inadditiontothemainpartofthepaper,weappendanelementaryalgebraicdiscussionoftheconnectionoftheboundaryvalueproblemofellipticequationswiththerandomwalkproblemarisinginstatistics.versity.*NowatCourantInstituteofMathematicalSciences,NewYorkUni-‘I’NowatUniversityofCalifornia,Berkeley.J.leRoux,“SurleproblemdeDirichlet”.Journ.demathCm.pur.etappl.(6)10,189(1914).R.G.D.Richardson,“Anewmethodinboundaryproblemsfordifferentialequations”,Trans.oftheAm.Math.SOC.18,p.489ff,(1917).(1925).UnfortunatelythesepaperswerenotknownbythefirstofthethreeH.B.PhilipsandN.Wiener,NetsandtheDirichletProblem,Publ.ofM.I.T.authorswhenhepreparedhisnote“Onthetheoryofpartialdifferenceequa-alsoL.Lusternik,“Onanapplicationofthedirectmethodinvariationcal-tions,”G6tt.Nachr.23,X,1925,onwhichthepresentworkdepends.Seeculus,”RecueildelaSocieteMathem.deMoscou1926.G.Bouligand,“SurleproblemmedeDirichlet,”Ann.delaSOC.polon.demathdm.4,Cracow(1926).Onthemeaningofthedifferenceexpressionsandonfurtherapplicationsofthem,seeR.Courant,“UberdirekteMethodeninderVariationsrechnung,”Math.Ann.07,p.711,andthereferencesgiventherein.215IBMJOURNALMARCH19671.TheellipticcaseSection1.Preliminaryremarks1.DefinitionsConsiderarectangulararrayofpointsinthe(x,y)-plane,suchthatformeshwidthh0thepointsofthelatticearegivenbyx=nhy=rnh1m,n=0,fl,f2,.LetGbearegionoftheplaneboundedbyacontinuousclosedcurvewhichhasnodoublepoints.Thenthecor-respondingmeshregion,G,-whichisuniquelydeter-minedforsufficientlysmallmeshwidth-consistsofallthosemeshpointslyinginGwhichcanbeconnectedtoanyothergivenpointinGbyaconnectedchainofmeshpoints.Byaconnectedchainofmeshpointswemeanasequenceofpointssuchthateachpointfollowsinthesequenceoneofitsfourneighboringpoints.WedenoteasaboundarypointofGhapointwhosefourneighboringpointsdonotallbelongtoG,.AllotherpointsofGhwecallinteriorpoints.Weshallconsiderfunctionsu,u,.ofpositiononthegrid,i.e.,functionswhicharedefinedonlyforgridpoints,butweshalldenotethemasu(x,y),u(x,y),..Fortheirforwardandbackwarddifferencequotientsweemploythefollowingnotation,u,=-[.(x+h,Y)-4x9Y)l,1hInthesamewaythedifferencequotientsofhigherorderareformed,e.g.,(UJ,=U,Z=Uzr1=2[.(x+h,Y-2u(x,Y+.(x-h,~11,etc.2.DifferenceexpressionsandGreen'sfunctionInordertostudylineardifferenceexpressionsofsecondorder,weform(usingasamodelthetheoryofpartialdifferentialequations),abilinearexpressionfromtheforwarddifferencequotientsoftwofun