5.3 Runge -Kutta Methods 183
f.Usetheanswers generated in(e)andpiecewise cubic Hermite interpolation toapproximate yat
thefollowing values andcompare them totheactual values ofy.
i. y(1.04) ii.>(1.55) iii.>(1.97)
9. Given theinitial -value problem
y'=}i-*-y2'i<»<2.:y(U=-i
with theexact solution >(/)=-1/f.
a.UseEuler ’smethod with h=0.05 toapproximate thesolution andcompare itwith theactual
values of>.
b.Usetheanswers generated in(a)andlinear interpolation toapproximate thefollowing values of
>andcompare them totheactual values .
i.>(1.052 ) ii.>(1.555 ) iii.>(1.978 )
c.UseTaylor ’smethod oforder 2with h=0.05 toapproximate thesolution andcompare itwith
theactual values of>.
d.Usetheanswers generated in(c)andlinear interpolation toapproximate thefollowing values of
>andcompare them totheactual values.
i.>(1.052 ) ii.>(1.555 ) iii.>(1.978 )
e.UseTaylor ’smethod oforder 4with h=0.05 toapproximate thesolution andcompare itwith
theactual values of>.
f.Usetheanswers generated in(e)andpiecewise cubic Hermite interpolation toapproximate the
following values of>andcompare them totheactual values .
i.>(1.052 ) ii.>(1.555 ) iii.>(1.978 )
10. Inanelectrical circuit with impressed voltage £,having resistance R,inductance L,andcapacitance
Cinparallel ,thecurrent isatisfies thedifferential equation
di d2£ 1d£ 1
dt~C~d^+R~
dt+L£'
Suppose i(0)=0,C=0.3farads ,R=1.4ohms ,L=1 . 7henries ,andthevoltage isgiven by
£(t)=eoab"sin(2f-jr).
UseEuler ’smethod tofindthecurrent iforthevalues t=0.1j,j=0,1,...,100.
11. Aprojectile ofmass m=0.11 kgshot vertically upward with initial velocity u(0)=8m/sisslowed
duetotheforce ofgravity FK=mgandduetoairresistance Fr=— w h e r e g=— 9.8m/s2
andk=0.002 kg/m.Thedifferential equation forthevelocity t;isgiven by
mu'=mg-&v|v|.
a.Find thevelocity after 0.1,0.2,...,1.0s.
b.Tothenearest tenth ofasecond ,determine when theprojectile reaches itsmaximum height and
begins falling.
5.3 Runge -Kutta Methods
Inthelastsection wesaw how Taylor methods ofarbitrary high order canbegenerated .
However ,theapplication ofthese high-order methods toaspecific problem iscomplicated
bytheneed todetermine andevaluate high-order derivatives with respect totontheright
side ofthedifferential equation .The widespread useofcomputer algebra systems has
simplified thisprocess ,butitstillremains cumbersome .
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Inthelater 1800 s,Carl Runge
(1856-1927 )used methods
similar tothose inthissection to
derive numerous formulas for
approximating thesolution to
initial-value problems .Inthissection weconsider Runge -Kutta methods ,which modify theTaylor methods so
thatthehigh-order error bounds arepreserved ,buttheneed todetermine andevaluate the
high-order partial derivatives iseliminated .The strategy behind these techniques involves
approximating aTaylor method withamethod thatiseasier toevaluate .This approximation
might increase theerror,buttheincrease does notexceed theorder ofthetruncation error
thatisalready present intheTaylor method .Asaconsequence ,thenewerror does not
significantly influence thecalculations .
In1901,Marlin Wilhelm Kutta
(1867-1944 )generalized the
methods thatRunge developed in
1895 toincorporate systems of
first-order differential equations .
These techniques differ slightly
from those wecurrently call
Runge -Kutta methods .Runge -Kutta Methods ofOrder Two
TheRunge -Kutta techniques make useoftheTaylor expansion of/,thefunction onthe
right side ofthedifferential equation .Since/isafunction oftwovariables ,tandy,we
must firstconsider thegeneralization ofTaylor ’sTheorem tofunctions ofthistype.This
generalization appears more complicated than thesingle -variable form ,butthisisonly
because ofallthepartial derivatives ofthefunction /.
Taylor 'sTheorem forTwo Variables
If/andallitspartial derivatives oforder lessthan orequal ton+1arecontinuous on
D={(f,y)|a<t<b,c<y<d]and(t,y)and(f+or,y+p)both belong toD,
then
/(/+<*,y+0)»/(f,y)+
a2d2f
2dt2r3/ a/
+32f B232f
(tJy )+a PW7jr(t,y)+~^j(t,y)313y 23y+
Theerror term inthisapproximation issimilar tothatgiven inTaylor ’sTheorem ,with
theadded complications thatarise because oftheincorporation ofallthepartial derivatives
oforder n+1.
Toillustrate theuseofthisformula indeveloping theRunge -Kutta methods ,letus
consider aRunge -Kutta method oforder 2.Wesawintheprevious section thattheTaylor
method oforder 2comes from
yft+i)=y(t.)+hy'(u)+y/'(»,)+y/"(£)
=y{ti)+hf(t„ym+y/'te,>(»,))+y/'tt),
or,since
/'(»,.yft))=y(l,))'
andy'fa)=/fe,y(t,)),wehave
yOi+i)=y(tj)+h/(»..yM )+~ (t„yM )+\yft)) fdi,y(t,))
h3
+3!y'"(?)-
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Taylor ’sTheorem oftwovariables permits ustoreplace theterm inthebraces with a
multiple ofafunction evaluation of/oftheform a\/(f,+a,y(fr)+P).Ifweexpand this
term using Taylor ’sTheorem with n=1,wehave
a x f {t i+ar,y(/,)+/3)%a,f(ti>yW )+ yfc))+ yfc))
df df= y(ti))+axa-^(tny(ti))+ y(tf)).
Equating thisexpression with theterms enclosed inthebraces inthepreceding equation
implies thatai,or,andpshould bechosen sothat
1=au -=a\a,and-/(r,,y(t,))=a\P\
thatis,
a\=1,a=
^,^0=x/ft.?(0)-
Theerror introduced byreplacing theterm intheTaylor method with itsapproximation
hasthesame order astheerror term forthemethod ,sotheRunge -Kutta method produced
inthisway,called theMidpoint method ,isalsoasecond -order method .Asaconsequence ,
thelocal error ofthemethod isproportional toA3,andtheglobal error isproportional {oh2.
Midpoint Method
w0=Of
W,+1=W,+h|/~,W/+ ,
where i=0,1,...,N—1,with local error0(hy)andglobal error0(h2).
Using a\f(t+a,y+p)toreplace theterm intheTaylor method istheeasiest choice ,
butitisnottheonly one.Ifweinstead useaterm oftheform
a\fit,y)+a2f(t+a,y+Pf(t,y)),
theextra parameter inthisformula provides aninfinite number ofsecond -order Runge -Kutta
formulas .When a\=a2=\anda=p=/z,wehave theModified Euler method .
Modified Euler Method
w0=a
Wi+,=w,-I--[/(/,,w,)+/(/,+1,w,-I-hfOi,W,))]
where i=0,1,...,N— 1,with local error0(h3)andglobal error0(h2).
Example 1UsetheMidpoint method andtheModified Euler method with N=10,h=0.2,t,=0.2*,
andwo=0.5toapproximate thesolution toourusual example ,
y'=y-t2+1,0<f<2,y(0)=0.5.
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Table 5.6
Karl Heun (1859-1929 )wasa
professor attheTechnical
University ofKarlsruhe .He
introduced thistechnique ina
paper published in1900 [Hcu].
Heun 'sMethodSolution Thedifference equations produced from thevarious formulas are
Midpoint method :w,+i=1.22w;— 0.0088 /2— 0.008 /+0.218 ;
Modified Euler method :wi+\— 1.22w,-0.0088 /2— 0.008 /+0.216 ,
foreach /=0,1,. ..,9.The firsttwosteps ofthese methods give
Midpoint method :w,=1.22(0.5)-0.0088 (0)2-0.008 (0)+0.218 =0.828 ;
Modified Euler method :wx=1.22(0.5)-0.0088 (0)2-0.008 (0)+0.216 =0.826 ,
and
Midpoint method :w2=1.22 (0.828 )-0.0088 (0.2)2-0.008 (0.2)+0.218
=1.21136 ;
Modified Euler method :w2=1.22(0.826 )-0.0088 (0.2)2-0.008 (0.2)+0.216
=1.20692 ,
Table 5.6listsalltheresults ofthecalculations .Forthisproblem ,theMidpoint method
issuperior totheModified Euler method .
t, yiOMidpoint
Method ErrorModified Euler
Method Error
0.0 0.5000000 0.5000000 0 0.5000000 0
0.2 0.8292986 0.8280000 0.0012986 0.8260000 0.0032986
0.4 1.2140877 1.2113600 0.0027277 1.2069200 0.0071677
0.6 1.6489406 1.6446592 0.0042814 1.6372424 0.0116982
0.8 2.1272295 2.1212842 0.0059453 2.1102357 0.0169938
1.0 2.6408591 2.6331668 0.0076923 2.6176876 0.0231715
1.2 3.1799415 3.1704634 0.0094781 3.1495789 0.0303627
1.4 3.7324000 3.7211654 0.0112346 3.6936862 0.0387138
1.6 4.2834838 4.2706218 0.0128620 4.2350972 0.0483866
1.8 4.8151763 4.8009586 0.0142177 4.7556185 0.0595577
2.0 5.3054720 5.2903695 0.0151025 5.2330546 0.0724173
Higher -Order Runge -Kutta Methods
The term T(3)(/,y)canbeapproximated with global error0{h?)byanexpression ofthe
form
fit+ ,y-I-<$if{t+a2,y+8 2fit,y))),
involving four parameters ,butthealgebra involved inthedetermination oforj,$1,ar2,and
<$2isquite involved .The most common 0(h*)isHeun ’smethod .
w0=a
W/+1= l(/(*i»W i)+3(/(t,+y,W,+y/(r,+|,w,+1/(*,,H',))))),
for1=0,1,. ..,N— 1,with local error0(h4)andglobal error0(h}).
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Illustration Applying Heun ’smethod with N=10,h=0.2,f,=0.2i,andwo=0.5toapproximate
thesolution toourusual example ,
y'=y-t2+1,0<t<2,y(0)=0.5
gives thevalues inTable 5.7.Note thedecreased error throughout therange over theMidpoint
andModified Euler approximations .
Table 5.7
ti y(ti)Heun ’s
Method Error
0.0 0.5000000 0.5000000 0
0.2 0.8292986 0.8292444 0.0000542
0.4 1.2140877 1.2139750 0.0001127
0.6 1.6489406 1.6487659 0.0001747
0.8 2.1272295 2.1269905 0.0002390
1.0 2.6408591 2.6405555 0.0003035
1.2 3.1799415 3.1795763 0.0003653
1.4 3.7324000 3.7319803 0.0004197
1.6 4.2834838 4.2830230 0.0004608
1.8 4.8151763 4.8146966 0.0004797
2.0 5.3054720 5.3050072 0.0004648
Theprogram RK04M52
implements the
Rungc -Kulta Order 4
method .Runge -Kutta Order Four
Runge -Kutta methods oforder 3arenotgenerally used .The most common Runge -Kutta
method inuseisoforder 4.Itisgiven bythefollowing .
Runge -Kutta Method ofOrder 4
w0=a,
*i=h f(t i,W j),
k2=hffti + Wi+
*3=A/(f,+£,W,+K),
*4= +k3).
w/+1=wi+ +2k2+2k$+£4),
where 1=0,1,.. .,N— 1,with local error0(/z5)andglobal error0(/i4).
Example 2Use theRunge -Kutta method oforder 4with h=0.2,N=10,andt,=0.2itoobtain
approximations tothesolution oftheinitial -value problem
y'=y-t2+1,0<r<2,y(0)=0.5 .
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Solution Theapproximation to>(0.2)isobtained by
wo=0.5
*i=0.2/(0,0.5)=0.2(1.5)=0.3
*2=0.2/(0.1,0.65 )=0.328
*3=0.2/(0.1,0.664 )=0.3308
*4=0.2/(0.2,0.8308 )=0.35816
w,=0.5+4(0.3+2(0.328 )+2(0.3308 )+0.35816 )=0.8292933 .6
Theremaining results andtheir errors arelisted inTable 5.8.
Table 5.8
tiExact
y<=y(t/)Runge -Kutta
Order 4
w,Error
1>t~w/l
0.0 0.5000000 0.5000000 0
0.2 0.8292986 0.8292933 0.0000053
0.4 1.2140877 1.2140762 0.0000114
0.6 1.6489406 1.6489220 0.0000186
0.8 2.1272295 2.1272027 0.0000269
1.0 2.6408591 2.6408227 0.0000364
1.2 3.1799415 3.1798942 0.0000474
1.4 3.7324000 3.7323401 0.0000599
1.6 4.2834838 4.2834095 0.0000743
1.8 4.8151763 4.8150857 0.0000906
2.0 5.3054720 5.3053630 0.0001089
Computational Comparisons
The main computational effort inapplying theRunge -Kutta methods involves thefunction
evaluations of/.Inthesecond -order methods ,thelocal error is0(h3)andthecost is
two functional evaluations perstep.The Runge -Kutta method oforder 4requires four
evaluations perstep andthelocal error is0(h5).Therelationship between thenumber of
evaluations perstepandtheorder oftheglobal error isshown inTable 5.9.Because ofthe
relative decrease intheorder forngreater than 4,themethods oforder lessthan 5with
smaller stepsizeareused inpreference tothehigher -order methods using alarger stepsize.
Table 5.9Evaluations perstep: 2 3 4 5<n<7 8<n<9 10<n
Best possible global error:o(/i2) O(A’)o(h*)0{hH~x) 0(hH~2)0{hn~3)
One way tocompare thelower -order Runge -Kutta methods isdescribed asfollows :
TheRunge -Kutta method oforder 4requires four evaluations perstep,sotobesuperior to
Euler ’smethod ,which requires only oneevaluation perstep,itshould givemore accurate
answers than when Euler ’smethod uses one-fourth theRunge -Kutta step size.Similarly ,
iftheRunge -Kutta method oforder 4istobesuperior tothesecond -order Runge -Kutta
methods ,which require twoevaluations perstep,itshould give more accuracy with step
sizehthan asecond -order method with stepsize\h.Thefollowing Illustration indicates
thesuperiority oftheRunge -Kutta method oforder 4bythismeasure .
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111ustrati on Fortheproblem
y'=y-t2+l,0<t<2,y(0)=0.5,
Euler ’smethod with h=0.025 ,theMidpoint method with h=0.05,andtheRunge -
Kutta fourth -order method with h=0.1arccompared atthecommon mesh points of
these methods 0.1,0.2,0.3,0.4,and0.5.Each ofthese techniques requires 20function
evaluations todetermine thevalues listed inTable 5.10 toapproximate,y(0.5).Inthis
example ,thefourth -order method isclearly superior .
Table 5.10
ti ExactEuler
h=0.025Modified
Euler
h=0.05Runge -Kutta
Order 4
h=0.1
0.0 0.5000000 0.5000000 0.5000000 0.5000000
0.1 0.6574145 0.6554982 0.6573085 0.6574144
0.2 0.8292986 0.8253385 0.8290778 0.8292983
0.3 1.0150706 1.0089334 1.0147254 1.0150701
0.4 1.2140877 1.2056345 1.2136079 1.2140869
0.5 1.4256394 1.4147264 1.4250141 1.4256384
MATLAB uses methods toapproximate thesolutions toordinary differential equations
thataremore sophisticated than thestandard Runge -Kutta techniques .Anintroduction to
methods ofthistype isconsidered inSection 5.6.
E X E R C I S E S E T 5 . 3
1. Use theModified Euler method toapproximate thesolutions toeach ofthefollowing initial -value
problems ,andcompare theresults totheactual values .
a.y'— te*'-2y,0<t<1,y(0)=0,with /i=0.5;actual solution y(f)=Ife3/— ~e3’+T*"2' 25C*
b.y'=1+(f— y)2,2</<3,y(2)=1,with h=0.5;actual solution y(t)=/4-
c.y'=1+y/f,1<t<2,y(l)=2,with h=0.25;actual solution y(r)=tint+2t.
d.y'=cos2f+sin3/,0<t<1,y(0)=1,with h=0.25;actual solution y(f)=1sin2/-Icos3/+
2. UsetheModified Euler method toapproximate thesolutions toeach ofthefollowing initial-value
problems ,andcompare theresults totheactual values .
a.y'=y/t— (y/f)2,1<t<2,y(l)=1,withh=0.1;actual solution y(f)=//(1+In/).
b.y'— l+y/f+(y/02,1<f<3,y(l)=0,withh=0.2;actual solution y(/)=ftan(lnf).
c.y'=— (y4*l)(y+3),0</<2,y(0)=— 2,with h=0.2;actual solution y(f)=
— 3+2(14-e~2/)"1.
d.y'=— 5y4-5/2+2f,0<t<1,y(0)= with /?=0.1;actual solution y(t)=t2+\e~il.
3. Use theModified Euler method toapproximate thesolutions toeach ofthefollowing initial -value
problems ,andcompare theresults totheactual values .
22^,2^^|
a.y'=^ ^,0</<1,y(0)=1,with h=0.1;actual solution y(/)=^ ^.
V2 -1b.y'=,1<t<2,y(l)= with h=0.1;actual solution y(f)=— — .14-/ln~ ln(r+1)
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4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.c.y'=(y2+y)/f,1<t<3,y(l)=— 2,with h=0.2;actual solution y(/)=-— -.
d.y'=— t y+4t/y,0</<1,y(0)=1,with /i=0.1;actual solution y(t)=\A— 3e_,\
Repeat Exercise 1using theMidpoint method .
Repeat Exercise 2using theMidpoint method .
Repeat Exercise 3using theMidpoint method .
Repeat Exercise 1using Heun ’smethod .
Repeat Exercise 2using Heun ’smethod .
Repeat Exercise 3using Heun ’smethod .
Repeat Exercise 1using theRunge -Kutta method oforder 4.
Repeat Exercise 2using theRunge -Kutta method oforder 4.
Repeat Exercise 3using theRunge -Kutta method oforder 4.
Usetheresults ofExercise 2andlinear interpolation toapproximate values ofy(f),andcompare the
results totheactual values .
a.y(1.25)andy(1.93) b.y(2.1)andy(2.75)
c.y(1.3)andy(1.93 ) d.>(0.54)and>(0.94)
Usetheresults ofExercise 3andlinear interpolation toapproximate values ofy(f),andcompare the
results totheactual values .
a.>(0.54 )and>(0.94 ) b.>(1.25)and>(1.93)
c.>(1.3)and>(2.93 ) d.>(0.54)and>(0.94)
Use theresults ofExercise 11andCubic Hermite interpolation toapproximate values of>(/),and
compare theapproximations totheactual values ,
a.>(1.25 )and>(1.93 ) b.>(2.1)and>(2.75)
c.>(1.3)and>(1.93 ) d.>(0.54)and>(0.94)
Usetheresults ofExercise 12andCubic Hermite interpolation toapproximate values of>(f),and
compare theapproximations totheactual values ,
a.>(0.54 )and>(0.94 ) b.>(1.25)and>(1.93)
c.>(1.3)and>(2.93 ) d.>(0.54)and>(0.94)
Show thattheMidpoint method andtheModified Euler method give thesame approximations tothe
initial -value problem
>'=->+r+l,0<f<1,>(0)=1,
foranychoice ofh.Why isthistrue?
Water flows from aninverted conical tank with acircular orifice attherate
ft=
where ristheradius oftheorifice ,xistheheight oftheliquid level from thevertex ofthecone ,
and A(x)isthearea ofthecross-section ofthetank xunits above theorifice .Suppose r=0.1ft,
g=32.1 ft/s2,andthetank hasaninitial water level of8ftandinitial volume of512(7r/3)ft3.Use
theRunge -Kutta method oforder 4tofindthefollowing .
a.The water level after 10minwith h=20s
b.When thetank willbeempty ,towithin 1min.
Theirreversible chemical reaction inwhich twomolecules ofsolid potassium dichromate (KzC^O?),
twomolecules ofwater (H20),andthree atoms ofsolid sulfur (S)combine toyield three molecules of
thegassulfur dioxide (S02),four molecules ofsolid potassium hydroxide (KOH ),andtwomolecules
ofsolid chromic oxide (Cr203)canberepresented symbolically bythestoichiometric equation :
2K2Cr207+2H20+3S— >4KOH +2Cr2Oj+3SO..
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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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