5.4 Predictor -Corrector Methods 191
Ifn!molecules ofK2Cr207,n2molecules ofH20,andn3molecules ofSareoriginally available ,the
following differential equation describes theamount x(r)ofKOH after time t:
dx
dt
where kisthevelocity constant ofthereaction .Ifk=6.22 x10-19,nj=n2=2x103,and
=3x103,usetheRunge -Kutta method oforder 4todetermine how many units ofpotassium
hydroxide willhave been formed after 0.2s.
20. Show thatHeun’sMethod canbeexpressed indifference form,similar tothatoftheRunge -Kutta
method oforder 4,as
w0=or.
*2=hf
^t,+ w,+-*|J,
*3=h f +y,w,+ ,
w.+i=W,+\(k{+3*3),
foreachi=0,1,...,N-1.
5.4 Predictor -Corrector Methods
TheTaylor andRunge -Kutta methods areexamples ofone-stepmethods forapproximating
thesolution toinitial -value problems .These methods usew,intheapproximation w,+i
to butdonotinvolve anyoftheprior approximations WQ,w w,*_|.Generally
some functional evaluations of/arerequired atintermediate points ,butthese arediscarded
assoon asw,+iisobtained .
Since \y(tj)-Wj|decreases inaccuracy asjincreases ,better approximation methods
canbederived if,when approximating y(4+j),weinclude inthemethod some ofthe
approximations prior tow,.Methods developed using thisphilosophy arecalled multistep
methods .Inbrief ,one-step methods consider what occurred atonly oneprevious step;
multistep methods consider what happened atmore than oneprevious step.
Toderive amultistep method ,suppose thatthesolution totheinitial-value problem
dy—=fit,y),fora<t<b,with y{a)=a,at
isintegrated over theinterval [4,4+1].Then
/ */+1 ru+1
y(4+i)-y(ti)=/y\t)dt=/fit,y(t) )dt,
Jtj Jti
and
ru+1
y(4+i)=yM+/fit,yit))dt.
Ju
Since wecannot integrate /(/,yit) )without knowing yit),which isthesolution tothe
problem ,weinstead integrate aninterpolating polynomial ,Pit),forfit,yit))determined
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bysome ofthepreviously obtained data points (to,wo),ft,w\),...,(tj,w,).When we
assume ,inaddition ,thatyft)%w,,wehave
/*'I+I
yft+i)^w,+/P(t)dt.
J t j
Ifwm+jisthefirstapproximation generated bythemultistcp method ,then weneed to
supply starting values wo,w\,...,wmforthemethod .These starting values arcgenerated
using aone-step Runge -Kutta method with thesame error characteristics asthemultistep
method .
There aretwodistinct classes ofmultistep methods .Inanexplicit method ,w,+idoes
notinvolve thefunction evaluation /ft+j,w,+|).Amethod thatdoes depend inparton
/ft+i,w,+i)isanimplicit method .
Adams -Bashforth Explicit Methods
Some oftheexplicit multistep methods ,together with their required starting values and
local error terms ,aregiven next .
Adams -Bashforth Two -Step Explicit Method
w0=a,w!=o?i,
w,+1=W|+-[3/ft,W i)-/ft— 1,W|_,)],
where i=1,2,...,N— 1,with local error^y"'0z,)/i3forsome /Zjinft.j,fl+i).
Adams -Bashforth Three -Step Explicit Method
w0=a,wi=a i,w2=a2,
w,+i=w,+— [23/ft,w,)-16/ft_i,w,_i)+5/ft_2,w,_2)]
where i=2,3,...,N— 1,with local error|y(4)0z,)/i4forsome /z(inft_2ff,+i).
Adams -Bashforth Four -Step Explicit Method
w0=a,w\=ori,w2=a2,w3=a3,
Wf+1=Wi+— [55/ft,w.)-59/ft_i,w,_i)+37/ft_2.w,_2)-9/ft_3>w,_3)]
where i=3,4,...,N— 1,with local error^y(5)(/z,)/t5forsome jz,inft_3,/,+i).
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Adams -Bashforth Five -Step Explicit Method
Wo=or,W!=a,,w2=a2,w3=or3,w4=a4
w,+i=w,+^[1901/ft.w,)-2774/ft_i,wl_i)
+2616/(f,_2,w,_2)-1274/(f i-3,w,_3)+251/(fl_4,w,_4)]
where i=4,5,.. .,N— 1,with local error^y(6,(/ij)h6forsome /x,-in(f,_4,f,+i).
Adams -Moulton Implicit Methods
Implicit methods use(f,+i,f(t,+1,wJ+i))asanadditional interpolation node intheapprox -
imation oftheintegral
r*+i
f(t,y(t))dt.
Some ofthemore common implicit methods arelisted next .Notice thatthelocal error
ofan(m-l)-step implicit method is0(/im+I),thesame asthatofanm-step explicit
method .They both usemfunction evaluations ,however ,because theimplicit methods use
/(f,+1,wl+i),buttheexplicit methods donot.
Adams -Moulton Two -Step Implicit Method
w0=a,wi=ofi
w,+i=Wi-h~[5/(fj+i,w1+I)+8/(4,w/)— w,_i)]
where i=1,2,. ..,N-1,with local error-^y(4)(M,)/z4forsome /x,in f/+i).
Adams -Moulton Three -Step Implicit Method
w0=a,w i=a uw2=a2,
w,+i=W,— [9/ft+i,wt+1)-f19/(r/,w,)-5/tt_
i,Wi_i)+/(f,_
2,w,_
2)],24
where i=2,3,.. .,N— 1,with local error-^y(5>(/x,)/i5forsome /xfin(/,_2,f,+1).
Adams -Moulton Four -Step Implicit Method
w0=Of,Wi=or,,w2=a2,w3=a3,
W,-+1=W/+— [251/fc+i,w,+1)+646/(r,,wf)-264/(r,_,,w,_i)
+106/(f,_
2,w,_
2)-19/(f,_
3,w/_
3)]
where i=3,4,.. .,N— 1,with local error-j^jy(6>0x,)/z6forsome jxfin(/,_3,r1+i).
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Itisinteresting tocompare anm-step Adams -Bashforth explicit method toan(m— 1)-
step Adams -Moulton implicit method .Both require mevaluations of/perstep,andboth
have theterms y(m+l)(iii)hm+xintheir local errors.Ingeneral ,thecoefficients oftheterms
involving /intheapproximation andthose inthelocal error aresmaller fortheimplicit
methods thanfortheexplicit methods .This leads tosmaller truncation andround -offerrors
fortheimplicit methods .
Example 1InExample 2ofSection 5.3(seeTable 5.8onpage 188)weused theRunge -Kutta method
oforder 4with h=0.2toapproximate thesolutions totheinitial value problem
y'=y-t2+1,0<f<2,y(0)=0.5.
The firstapproximations were found tobey(0)=WQ=0.5,>(0.2) W\=0.8292933 ,
>»(0.4)%W2=1.2140762 ,andy(0.6)^w$=1.6489220 .Usethese asstarting values
forthefourth -order Adams -Bashforth method tocompute new approximations fory(0.8)
and>(1.0),andcompare these newapproximations tothose produced bytheRunge -Kutta
method oforder 4.
Solution Forthefourth -order Adams -Bashforth wehave
y(0.8)w4=w3+!TT(55/(0.6,w3)-59/(0.4,w2)+37/(0.2,w,)-9/(0,w0))24
=1.6489220 +— (55/(0.6,1.6489220 )-59/(0.4,1.2140762 )24
-I-37/(0.2,0.8292933 )-9/(0,0.5))
=1.6489220 +0.0083333 (55(2.2889220 )-59(2.0540762 )
+37(1.7892933 )— 9(1.5))
=2.1272892 ,
and
y(1.0)**w5=w4+^(55/(0.8,w4)-59/(0.6,w3)+37/(0.4,w2)-9/(0.2,w,))24
=2.1272892 +^(55/(0.8,2.1272892 )-59/(0.6,1.6489220 )24
+37/(0.4,1.2140762 )-9/(0.2,0.8292933 ))
=2.1272892 +0.0083333 (55(2.4872892 )-59(2.2889220 )
+37(2.0540762 )-9(1.7892933 ))
=2.6410533 ,
Theerrors forthese approximations att=0.8andt=1.0are,respectively ,
12.1272295 -2.12728921 =5.97 x10~5and12.6410533 -2.64085911 =1.94 x10~“ .
Thecorresponding Runge -Kutta approximations haderrors
12.1272027 -2.12728921 =2.69x10“ 5and|2.6408227 -2.64085911 =3.64 x10“ 5.
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Theprogram PRCORM 53
implements theAdams
Predictor -Corrector
method .
Example 2Theimplicit Adams -Moulton methods generally give considerably better results than
theexplicit Adams -Bashforth method ofthesame order.However ,theimplicit methods
have theinherent weakness offirsthaving toconvert themethod algebraically toanexplicit
representation forwJ+i.That thisprocedure canbecome difficult ,ifnotimpossible ,canbe
seen byconsidering theelementary initial -value problem
y'=ey\ for0<t<0.25,with y(0)=1.
Since/(/,y)=e>\theAdams -Moulton Three -Step method has
w1+1=w,+^-[9e‘',+1+196“ '-5ew‘-'+ew-2]24
asitsdifference equation ,andthisequation cannot besolved explicitly forw,-+j.Wecould
useNewton ’smethod ortheSecant method toapproximate w,+i,butthiscomplicates the
procedure considerably .
Predictor -Corrector Methods
Inpractice ,implicit multistep methods arenotused alone .Rather ,they areused toim-
prove approximations obtained byexplicit methods .The combination ofanexplicit and
implicit technique iscalled apredictor -corrector method .The explicit method predicts
anapproximation ,andtheimplicit method corrects thisprediction .
Consider thefollowing fourth -order method forsolving aninitial -value problem .The
first step istocalculate thestarting values wo,wj,w2,and w3fortheexplicit Adams -
Bashforth Four-Step method .Todothis,weuseafourth -order one-stepmethod ,specifically ,
theRungc -Kutta method oforder 4.The next step istocalculate anapproximation ,w4p,to
y(f4)using theexplicit Adams -Bashforth Four-Step method aspredictor :
hw4p=w3+^[55/(f3,w3)-59/(r2,w2)+37/(f,,w,)-9/(f0,vv0)].
This approximation isimproved byuseoftheimplicit Adams -Moulton Three -Step method
ascorrector :
W4=w3+-[9/(/4,w4p)-f19/(r3,w3)-5/(r2,w2)+/(/1,w\)].
The value w4isnow used astheapproximation toy(r4).Then thetechnique ofusing the
Adams -Bashforth method asapredictor andtheAdams -Moulton method asacorrector is
repeated tofindw$pandw3,theinitial andfinal approximations toy(ts).This process is
continued until weobtain anapproximation toy(tjv)=y(b).
Program PRCORM 53isbased ontheAdams -Bashforth Four-Step method aspredictor
andoneiteration oftheAdams -Moulton Three -Step method ascorrector ,with thestarting
values obtained from theRunge -Kutta method oforder 4.
Apply theAdams fourth -order predictor -corrector method with h=0.2andstarting values
from theRunge -Kutta fourth -order method totheinitial -value problem
y'=y-t2+1,0<t<2,y(0)=0.5.
Solution This isacontinuation andmodification oftheproblem considered inExample 1
atthebeginning ofthesection .Inthatexample ,wefound thatthestarting approximations
from Runge -Kutta arc
y(0)=w0=0.5,y(0.2)%w,=0.8292933 ,y(0.4)%w2=1.2140762 ,and
y(0.6)%w3=1.6489220 .
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andthefourth -order Adams -Bashforth method gave
>(0.8)%w4p=Wj+°'2[55/(0.6,w3)-59/(0.4,w2)+37/(0.2,w,)-9/(0,tv0)]24
=1.6489220 +^[55/(0.6,1.6489220 )-59/(0.4,1.2140762 )24
+37/(0.2,0.8292933 )-9/(0,0.5)]
=1.6489220 +0.0083333 [55(2.2889220 )-59(2.0540762 )
+37(1.7892933 )— 9(1.5)]
=2.1272892 .
Wewill now usew4pasthepredictor oftheapproximation toy(0.8)anddetermine the
corrected value W4,from theimplicit Adams -Moulton method .This gives
y(0.8)^W4:=W3H-^[9/(0.8,w4p)4-19/(0.6,w3)-5/(0.4,w2)4-/(0.2,vvO]
=1.6489220 4-— [9/(0.8,2.1272892 )4-19/(0.6,1.6489220 )24-5/(0.4,1.2140762 )4-/(0.2,0.8292933 )]
=1.6489220 +0.0083333 (9(2.4872892 )4-19(2.2889220 )
-5(2.0540762 )+(1.7892933 )]
=2.1272056 .
Now weusethisapproximation todetermine thepredictor ,w5/„ fory(1.0)as
V;o is
IIIIIIII=+^[55/(0.8,w4)-59/(0.6,w3)+37/(0.4,w2)-9/(0.2,w,)]
=2.1272056 +°'2[55/(0.8,2.1272056 )-59/(0.6,1.6489220 )
+37/(0.4,1.2140762 )-9/(0.2,0.8292933 )]
=2.1272056 4-0.0083333 (55(2.4872056 )-59(2.2889220 )
4-37(2.0540762 )-9(1.7892933 )]
=2.6409314 ,
andcorrect thiswith
y(1.0)%W5:=W«+^[9/d.O,W ip )+19/(0.8,w4)-5/(0.6,H-3)+/(0.4,w2)]
=2.1272056 +— [9/(1.0,2.6409314 )+19/(0.8,2.1272892 )24
-5/(0.6,1.6489220 )4-/(0.4,1.2140762 )]
=2.1272056 4-0.0083333 (9(2.6409314 )-1-19(2.4872056 )
-5(2.2889220 )4-(2.0540762 )]
=2.6408286 .
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InExample 1wefound thatusing theexplicit Adams -Bashforth method alone produced
results thatwere inferior tothose ofRunge -Kutta.However ,these approximations to>(0.8)
and>(1.0)areaccurate towithin
12.1272295 -2.12720561 =2.39x10” 5and
12.6408286 -2.64085911 =3.05xllT5,
respectively ,compared tothose ofRunge -Kutta ,which were accurate ,respectively ,to
within
12.1272027 -2.1272892 |=2.69 x1(T5and
12.6408227 -2.6408591 |=3.64x10"5.
Other multistep methods canbederived using integration ofinterpolating polynomials
over intervals oftheform [tj,r,+i]forj<i— 1,where some ofthedata points areomitted .
Milne ’smethod isanexplicit technique thatresults when aNewton Backward -Difference
interpolating polynomial isintegrated over[#, _3,
Milne 'sMethod
Wf+l=W,_
3+y[2/fo,W i)-/fo-1,W,_,)+2/(r,_
2,wl_
2)],
where i=3,4,...,N— 1,with local error||/I5><5)(M,)forsome /x,in(f,_3,/,+1).
This method isused asapredictor foranimplicit method called Simpson ’smethod .
Itsname comes from thefactthatitcanbederived using Simpson ’sruleforapproximating
integrals .
Simpson 'sMethod
W|+1=Wj-i+j[/(fj+i*wj+i)+4/(f|,w,-)+/ft-ifw#_
i)],
where 1'=1,2,...,N— 1,with local error — ^A5>(5)(Mi)forsome /x,in(/, _1,f,+i).
Although thelocal error involved with apredictor -corrector method oftheMilne -
Simpson type isgenerally smaller than thatoftheAdams -Bashforth -Moulton method ,the
technique haslimited usebecause ofround -offerror problems ,which donotoccur with the
Adams procedure .
MATLAB uses methods that aremore sophisticated than thestandard Adams -
Bashforth -Moulton techniques toapproximate thesolutions toordinary differential equa-
tions.Anintroduction tomethods ofthistype isconsidered inSection 5.6.
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EXERCISE SET 5.4
1. Use alltheAdams -Bashforth methods toapproximate thesolutions tothefollowing initial -value
problems .Ineach case,useexact starting values andcompare theresults totheactual values .
a.y'=te2‘— 2y,for0</<1,with y(0)=0andh=0.2;actual solution y(t)=Jte3>— ± eyt+± e-2t
25e*25r
b.y'=l+(r— y)2,for2<t<3,withy (2)=1and/i=0.2;actual solution y(t)=t-1-1/(1— /).
c.y=1+y,for1<t<2,with y(l)=2andh=0.2;actual solution y(f)=tInt+It.
d.y=cos2/+sin3f,for0</<1with y(0)=1andh=0.2;actual solution y(f)=
Isin2r-|cos3f+
2. UsealltheAdams -Moulton methods toapproximate thesolutions toExercises 1(a),1(c),and1(d).
Ineach case ,useexact starting values andexplicitly solve for .Compare theresults totheactual
values .
3. Useeach oftheAdams -Bashforth methods toapproximate thesolutions tothefollowing initial -value
problems .Ineach case ,usestarting values obtained from theRunge -Kutta method oforder 4.Compare
theresults totheactual values .
a.y'=j— ,for1<r<2,withy (l)=1andh=0.1;actual solution y(r)=r/(l+lnr).
b./=l+~~+.for1<t<3,withy (l)=Oand /i=0.2;actual solution y(r)=ttan(lnr).
c.y=-(y+l)(y+3),for0<t<2,with y(0)=-2andh=0.1;actual solution
y(0=— 3+2/(14-e-2/).
d.y'=— 5y+512+2/,for0<t<1,with y(0)=1/3andh=0.1;actual solution
y(t)=t2+\e~u.
4. Use thepredictor -corrector method based ontheAdams -Bashforth Four-Step method and the
Adams -Moulton Three -Step method toapproximate thesolutions totheinitial -value problems in
Exercise 1.
5. Usethepredictor -corrector method based ontheAdams -Bashforth Four-Step method andtheAdams -
Moulton Three -Step method toapproximate thesolutions totheinitial -value problem inExercise 3.
6. Theinitial -value problem
y— ey,for0<t<0.20 ,with y(0)=1
hastheexact solution
y(t)=1— ln(l-et).
Applying theAdams -Moulton Three -Step method totheproblem requires solving forw(+1inthe
equation
g(M>i+i)=n-(+i-(H-,+^(9e"*1+19ew‘-5«-'-'+ =0.
Suppose thath— 0.01 andweusetheexact starting values forvv0,wi,andwi.
a.Apply Newton ’smethod tothisequation with thestarting value witoapproximate W3towithin
10-6.
b.Repeat thecalculations in(a)using thestarting value todetermine approximations accurate
towithin 10~6foreach w(>,for 1=2,...,19.
7. UsetheMilne -Simpson Predictor -Corrector method toapproximate thesolutions totheinitial -value
problems inExercise 3.
5.5Extrapolation Methods
Extrapolation wasused inRomberg integration fortheapproximation ofdefinite integrals ,
where wefound that,bycorrectly averaging relatively inaccurate trapezoidal approxima -
tions,wecould produce newapproximations thatareexceedingly accurate .Inthissection
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