They are motivated by the dependence of the Taylor methods on the speci64257c IVP These new methods do not require derivatives of the righthand side function in the code and are therefore generalpurpose initial value problem solvers RungeKutta metho ID: 27123
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2f(t;y)+h ExplicitRunge-Kuttamethodsarecharacterizedbyastrictlylowertriangularma-trixA,i.e.,aij=0ifji.Moreover,thecoecientsciandaijareconnectedbytheconditionci=Xj=1aij;i=1;2;:::;:Thissaysthatciistherowsumofthei-throwofthematrixA.Thisconditionisrequiredtohaveamethodoforderone,i.e.,forconsistency.Welimitourdiscussiontosuchmethodsnow.Thus,foranexplicitsecond-ordermethodwenecessarilyhavea11=a12=a22=c1=0.Wecannowstudywhatothercombinationsofb1,b2,c2anda21in(45)giveusasecond-ordermethod.ThebivariateTaylorexpansionyieldsf(t+c2h;y+ha21~k1)=f(t;y)+c2hft(t;y)+ha21fy(t;y)~k1+O(h2)=f(t;y)+c2hft(t;y)+ha21fy(t;y)f(t;y)+O(h2):Therefore,thegeneralsecond-orderRunge-Kuttaassumption(45)becomesy(t+h)=y(t)+h[b1f(t;y)+b2ff(t;y)+c2hft(t;y)+ha21fy(t;y)f(t;y)g]+O(h3)=y(t)+(b1+b2)hf(t;y)+b2h2[c2ft(t;y)+a21fy(t;y)f(t;y)]+O(h3):InorderforthistomatchthegeneralTaylorexpansion(43)wewantb1+b2=1c2b2=1 2a21b2=1 2:Thus,wehaveasystemofthreenonlinearequationsforourfourunknowns.Onepopularsolutionisthechoiceb1=0,b2=1,andc2=a21=1 2.ThisleadstothemodiedEulermethod(sometimesalsoreferredtoasthemidpointrule,seethediscussioninSection3.3below)yn+1=yn+hk2withk1=f(tn;yn)k2=f(tn+h 2;yn+h 2k1):ItsButchertableauxisoftheform 0 001 2 1 20 01. Remark Thechoiceb1=1,b2=0leadstoEuler'smethod.However,sincenowc2b26=1 2anda21b26=1 2thismethoddoesnothavesecond-orderaccuracy. 58 evaluationsoffpertimestep 2 3 4 5 6 7 8 9 10 11 maximumorderachievable 2 3 4 4 5 6 6 7 7 8 Table4:EciencyofRunge-Kuttamethods. Thesedataimplythathigher-order(�4)Runge-Kuttamethodsarerelativelyinecient.Precisedataforhigher-ordermethodsdoesnotseemtobeknown.However,certainhigher-ordermethodsmaystillbeappropriateifwewanttoconstructaRunge-Kuttamethodwhichadaptivelychoosesthestepsizeforthetimestepinordertokeepthelocaltruncationerrorsmall(seeSection5).3.3ConnectiontoNumericalIntegrationRulesWenowillustratetheconnectionofRunge-Kuttamethodstonumericalintegrationrules.Asbefore,weconsidertheIVPy0(t)=f(t;y(t))y(t0)=y0andintegratebothsidesofthedi erentialequationfromttot+htoobtainy(t+h)�y(t)=Zt+htf( ;y( ))d :(47)Therefore,thesolutiontoourIVPcanbeobtainedbysolvingtheintegralequation(47).Ofcourse,wecanusenumericalintegrationtodothis: 1. UsingtheleftendpointmethodZbaf(x)dxb�a n| {z }=hn�1Xi=0f(xi)onasingleinterval,i.e.,withn=1,anda=t,b=t+hwegetZt+htf( ;y( ))d t+h�t 1f( 0;y( 0))=hf(t;y(t))since 0=t,theleftendpointoftheinterval.Thus,aswesawearlier,(47)isequivalenttoEuler'smethod. 2. UsingthetrapezoidalruleZbaf(x)dxb�a 2[f(a)+f(b)]witha=tandb=t+hgivesusZt+htf( ;y( ))d h 2[f(t;y(t))+f(t+h;y(t+h))]: 60 4. Simpson'sruleyieldsthefourth-orderRunge-Kuttamethodincasethereisnodependenceoffony. 5. Gaussquadratureleadstoso-calledGauss-Runge-KuttaorGauss-Legendremeth-ods.Onesuchmethodistheimplicitmidpointruleyn+1=yn+hf(tn+h 2;1 2(yn+yn+1))encounteredearlier.TheButchertableauxforthisone-stageordertwomethodisgivenby 1 2 1 2 1. NotethatthegeneralimplicitRunge-Kuttamethodisoftheformyn+1=yn+hXj=1bjkjwithkj=f(tn+cjh;yn+hjXi=1aj;iki)forallvaluesofj=1;:::;.Thus,theimplicitmidpointrulecorrespondstoyn+1=yn+hk1withk1=f(tn+h 2;yn+h 2k1)|obviouslyanimplicitmethod. 6. MoregeneralimplicitRunge-Kuttamethodsexist.However,theirconstructionismoredicult,andcansometimesbelinkedtocollocationmethods.SomedetailsaregivenattheendofChapter3intheIserlesbook. 62