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1、CHAPTER26:STIFFNESSANDMULTISTEPMETHODSThechaptercovers:-1)stiffODEs(individualandsystem)(2)multistepmethods26.1Stiffness·Astiffsystem–rapidlychangingcomponentstogetherwithslowlychangingones.·Inmanycases,theslowlyvaryingcomponentswilldominatethesolutionwhichcausetherap
2、idlyvaryingcomponentsdieawayquickly.·Althoughthetransientphenomenaexistforonlyashortpartoftheintegrationinterval,theycandictatethetimestepfortheentiresolution.Example:dy/dt=-1000y+3000-2000e-twhereify(0)=0,thesolution,y=3-0.998e-1000t-2.002e-t(figure26.1)-thesolutioni
3、sinitiallydominatedbyfastexponentialterm(e-1000t),thenaftert<0.005,thistransientdiesoutandtheslowexponential(e-t)dominated.MATB324,June2001Chapter26-11HanimSalleh·Insightintothestepsizerequiredforstabilityofsuchsolutioncanbegainedbyexaminingthehomogeneouspart…dy/dt=-a
4、yIfy(0)=yothentheanalyticalsolution,y=y0e-atThusthesolutionstartsatyoandasymtoticallyapproacheszero.·UsingEuler’smethod,--ornotethatifh>2/a,
5、yi
6、à¥asIà¥.Or
7、1-ah
8、must<1.·Forthefasttransientcaseabove,thestepsizetomaintainstabilitymustbe<2/1000=0.002·Thiscriterionmaintain
9、sstability,howevertoobtainanaccuratesolution,asmallerstepsizewouldberequired.MATB324,June2001Chapter26-11HanimSalleh·Thus,althoughthetransientoccursforonlyasmallfractionoftheintegrationinterval,itcontrolsthemaximumallowablestepsize.·Oneremedyistousetheimplicitmethod.(
10、sincetheunknownsappearsonbothsidesoftheequations)·ForEuler’simplicitform(orbackwardform),itisdevelopedbyevaluatingthederivativeatfuturetime,Notethatasià¥,
11、yi
12、à0,thusitisunconditionallystable.Example26.1.·SystemsofODEscanalsobestiff,forexample,fortheinitialconditionsy1
13、(0)=52.29andy2(0)=83.82,theexactsolutionisMATB324,June2001Chapter26-11HanimSalleh·Theexponentsarenegativeanddifferbyabout2ordersofmagnitude.·AnimplicitEuler’smethodforsystemscanbeformulatedforthepresentexampleas·Collectingterms,andsolvefory1,i+1andy2,i+1foreachtimeste
14、p.MATB324,June2001Chapter26-11HanimSalleh26.2MultistepMethods.·Themultistepmethodsareanalternativeapproacheswhichbasedonthea
