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1、VOL.38,1952MATHEMATICS:CURTISSANDHIRSCHFELDER23510Proc.Roy.Physiog.Soc.Lund.,20,250-253(1950).11Comm.onPureandAppliedMath.,v.3(1950).12Courant-Hilbert,v.2,pp.488-490.13ActaMath.,41(1918).INTEGRATIONOFSTIFFEQUATIONS*ByC.F.CURTISSANDJ.0.HIRSCHFELDERTHENAVALRESEARCHLAB
2、ORATORY,DEPARTMENTOFCHEMISTRY,UNIVERSITYOFWISCONSIN,MADISON,WISCONSINCommunicatedbyFarringtonDaniels,December29,1951Inthestudyofchemicalkinetics,electricalcircuittheory,andproblemsofmissileguidanceatypeofdifferentialequationariseswhichisexceed-inglydifficulttosolveb
3、yordinarynumericalprocedures.Averysatis-factorymethodofsolution-oftheseequationsisobtainedbymakinguseofaforwardinterpolationprocess.Thisschemehastheunusualpropertyofsinglingoutandapproximatingaparticularsolutionofthedifferentialequationtotheexclusionofthemanifoldofo
4、thersolutions.Thisbehaviormaybeexplainedbyasimplegeometricalinterpretationofthesignificanceoftheforwardinterpolationprocess.Thedifferentialequationstowhichthismethodappliesarecalled"stiff."Atypicalexampleofastiffequationistheequationrepresentingtherateofformationoff
5、reeradicalsinacomplexchemicalreaction.Thefreeradicalsarecreatedanddestroyedsorapidlycomparedtothetimescalefortheover-allreactionthattoafirstapproximationtherateofproductionisequaltotherateofdepletion.Thisisthenotionofthepseudo-stationarystate.Insomecasessuchasthefas
6、treactionsoccurringinflamesordetonations,thisapproximationisnotsufficientlyaccurate.Themethoddescribedinthepresentpaperprovidesameansforobtainingsolutionstoequationsofthistypetoanydegreeofaccuracy.Thenunmericalproceduredescribedherecaneasilybeextendedtosetsofsimulta
7、neousfirst-orderdifferentialequations.Inanyparticularregion,thedifferentialequationscanbeuncoupledbyintroducingsuitablelinearcombinationsoftheoriginaldependentvariables.Someoftheuncoupledequationsmaybe"stiff"inwhichcasetheycanbeintegratedbythemethodsdiscussedhere;wh
8、ereasotheruncoupledequationsmaybeintegratedbythemoreusualprocedures.1.ConceptofStiffEquations.-Considerthefirst-orderdifferentialequation,
