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ECOM-5134JUNE 1967 AD

00to

N U M E R I C A L I N T E G R A T I O N M E T H O D S F O R B A L L I S T I C R O C K E T T R A J E C T O R Y S I M U L A T I O N

P R O G R A M S By Randall.alters

D D C \pr^r~nrr " ~ < O r

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NUMERICALINTEGRATION METHOPSFORBALLISTICROCKETTRAJECTORY SIMULATIONPROGRAMS

ByRANDALL K .WALTERS

ECOM -5134 June1967

DATASKIV014501BS3A-10

ATMOSPHERICSCIENCES LABORATORYWHITE SANDSMISSILERANHE,NEWMEXICO

Distribution of thisreporti s unlimited.

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*

ABSTRACT

Numericalintegrationmethodsfor solutionofthesystemofdifferentialequationsfoundin ballistic rocket trajectoryprogramsarediscussed.Thegeneral discussion entailstheexplicitformulasofRunge-Kutta andpredictor-corrector methods andtheir errors,and a briefdescriptionofothermethodsthatcouldbeemployed.

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CONTENTS

PageA BST RACT iiINTROLXTIONRUNGE-KUTTAMET HO D S

FormulasErrorAnalysis

MULTISTEP METHODSStandardPr e d i ct o r - Co r r e ct o rM e t h o d sTh eGeneralizedPr e d i ct o r - Co r r e ct o r 2A n al ys i sofPr e d i ct o r - Co r r e ct o r Methods 6

OTHERGENERAL MET HO D SBlockMethod7Hybrid Method 8

SUMMARY 8REFERENCES 0

APPENDIXA 2

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INTRODUCTION

I n a ballistic rocket trajectory simulation programthesystemof differentialequationsusedtodescribe theballisticmodeli sahighlycomplexsystem.n particular thesix-degree offreedommodelusedbyheAtmospheric SciencesOfficeat WhiteSands MissileRange,consistsof a systemof twenty-one 2ndorder ordinarydifferen-tialequations whichare tobe solvedfortheballistic rocket's com-ponentsofacceleration,velocity,andpositionatdiscretetimein-tervals.hemostfeasible methodforsolvingth esystem i sto pro-grami tfor a computer andsolve bysomenumericalintegrationorocedure.This report will presentseveralnumericalintegration schemeswhicharecurrently being usedorarefeasible foruse in a ballisticrocketsimulationprogram.*

RUNGE-K UTTA METHODSFormulas

Considerthe systemoffirst orderdifferentialequationsdyiy[ fA(x ,yjtx),y2(x),. . . ,yN( x)) ( 1 )

satisfying theinitialconditions

y.(x)* y . 2 )iv o ' 7i oW e desirethevaluesofy.(xh ) ,i1 ,2 ,. . . ,N ,where hi stheincrementintheindependentariable.*Thisreport was presented tothe SecondAnnual ConferenceonPureandAppliedMathematics held at New MexicoInstituteofMiningandTechnology,SocorroNew Mexico,F e b .24-25,1967.

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T h eRunge-Kutt a m e t h o disanalgorithm designedtoapproximateth e T a y l o rseriesso lutio n

y.(xh)y .hy .(}(x) ~.(2)(x) i^o 720 7i *o 2 7i *oJ

F y*- ( 3 )f k ) of(1),w h e r ey .(x)denotesthektherivativean di=1 ,2 ,

...,N.owever,inderivingthedefiningequationsofthealgorithm,theuseoftheappropriateassumptionsmakesitunnecessarytoeval-uateanddefinethederivativesof higherorder.

ThegeneralequationsdefiningafourthorderRunge-Kuttamethodare givenby

K ih

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On esolutionisthec l a s s i c a lformulasof Runge,given by,

K ,- h f ( x v)O''0'K 2 f (xo ,o jt)K 3 hf < X o4Vr> ( 5 )K4=hf(xo h,yot K3)K (Kj*2K2*2K3+K4)

Onewillnoticethatwheny '=f(x),Runge ' sfor mulasreducetoSimp-son'srule:

x*hK =/ f(x)dx= 2 .[f(xo) 4 f(x r t+ y)*f(xn h>]

0\ >T2J ( 6 )

Anothersolutiontotheonep a r a m e t e rfamil yisth ecl assicalformulasofKuttagivenby

K , * hfKYJ''o'K 2 h f ( x o ,y Q * j i j

2h *1 *2 ( 7 )K.h f (x h,yK .-K- K.)Kj( K j+3K2 3K3 K4)

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To applytheRunge-Kuttamethod using a highspeeddigitalcom-puter.Gill[ 8 ]developeda calculationprocedurewhich( a )equiresaminimumnumberof storage registers,( b )ives a highaccuracy,( c )equirescomparativelyfew instructions.

Gillaccomplishedthe abovebydefining anauxiliaryse t ofpa-rametersi n theinitialsolutions e t ,therebyobtainingth efollowingsolution;Kih

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where the Q .are the auxiliaryparameters.nitially0szerojto compensatefor the round-offerror in y . . Q , ,i s used a sth eQnthenexts t e p .or a completediscussionofthealgorithm andaflowchart,referto[ 1 6 ] .

Error

Toobtain^a generalexpressionfor truncationerrorper step,thefunctionsK = y(xh )- vndK *y(xh )- yreexpandedthroughterms of h,thusgivingtheper-steptruncationerror as

Es -K *mo-r(S2T ST2 f2y+ 3f 2fST - 4 f3 T ) 2fT3] . . . ( 9 )yy y J

whereTK = DKf ,SK = DKfy,P =( D f )2.

Lotkin[ 1 2 ]determined a boundforL whichi sgiven by| E |

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Anumericalintegrationmethod i s calledstableifatthenthstep,thetotalerror,whichi s bothtruncation andround-offerror,i s at a minimum*thereby forcing theapproximatesolutiontotendtothetrue solution.Hence,as forthe stability oftheRunge-kuttamethods,Carr[ 3 )provedatheorem,ofwhichtheessential parts are statedbelow:

Let fecontinuous,negative andboundedfrom aboveandbelowinsomeregionRofthe( x ,y )plane,i.e.,M2> M .>0 ,fyc(~M2,- M ) ;further,letE betheabsolutevalue ofthemaximumerror introducedateachstep,andD*be a regionin whichthe so-lution ofthe differenceequationtends tothey-boundary of DnocloserthanQh | e . j ,where

Qimax |f(x,y)| a x\A9\A |KJ]x,yeD2( 1 2 )ande .i s the errora ttheith step.hen,theKuttafourthordermethodhasaboundonthetotalerrorintheithstep,wherethisbound i s givenby

fei '-h M , ( 1 3 )

insomeregionD * ,andwherethestep-size,h ,i s givenbyM 4 M .3

h

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andihM

andEand hare as givenbefore.Statements( 1 3 ) ,( 1 4 ) ,( 1 5 )relatethestep-sizeandthe prop-agatederror.hus,an hcanbefoundthatwill make thepropagatederrorless thanacertainbound,ifthis bound i s theknownboundonthepartial derivative.lgorithms for finding such stepsizes

canbefound in Carr[ 3 ] .Although variousboundsonthetruncationerror areknown,suchas theonegivenabove,theirusage inacomputerprogram i s usuallynotfeasible.hus,somepracticalschemeforestimationof thiserror must bedevised.One suchschemewasdevisedbyRichardson[ 5 ] .hisschemei s basedonnumericallyintegratingwithstep-sizesofhandh . and

comparingtheresultsusingthesestep-sizes.variation ofthiswhichweusei sthe following:Let Ybethetruevalueofyat x+h, thevalueob-(2)tained a t x +husing h .h ,yv thevalue obtainedatx +h ( 1 6 )

using h9h hen fo r smallh ,72 )yTheschemein( 1 6 )canbeusedt o checkthevalidityof the answerforthepurposeofhalvinganddoublingthestep-sizeh .Analysis

By virtueoftheRunge-Kuttamethodsrequiringonlyinformationfromonepreviousstep,themethodhas desirablestability charac-teristicsandeaseofhalvingordoublingthestep-size h .

I thasbeenshown,however,byBlum[ 2 ]andFyfe[ 6 ] ,thatthroughproperredefinitionofthesolutions e t of parameters,mostotherversionsofthefourthorderRunge-KuttamethodcanavailthemselvesofthereducedstorageoftheGillalgorithm.

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I f t h ef i r s td e r i v a t i v e o f t h e f u n c t i o n sa r e v e r y i n v o l v e d , t h e ns e v e r a l e v a l u a t i o n so f t h e s ef i r s t d e r i v a t i v e s m a y b es o m e w h a t t i m ec o n s u m i n g a n d t h u s c a u s et h e m e t h o d t o b e u n e c o n o m i c a l a tt i m e s .A n o t h e r d i s t i n c t s h o r t c o m i n g o f t h e m e t h o d i st h a to f t h e e r r o r .N e i t h e rt r u n c a t i o n e r r o r ,n o r i t se s t i m a t e ,i so b t a i n e d i nt h ec a l - c u l a t i o n p r o c e d u r e , t h u s n e c e s s i t a t i n gt h e a p p r o x i m a t e c o m p a r i s o n sd e s c r i b e d p r e v i o u s l y .F i n a l l y , t h eR u n g e - K u t t a m e t h o d s c a n b e u s e d a sas t a r t i n g p r o -c e d u r e f o r o t h e r m e t h o d s ,s u c ha s m u l t i s t e p m e t h o d s .

j U L T I S T E P M E T H O D Si t a n d a r d P r e d i c t o r - C o r r e c t o r M e t h o d s

A m u l t i s t e p m e t h o d i sa m e t h o d i n w h i c h t h ec a l c u l a t i o n o f t h ey . * s a tt h e( M + l )s t s t e pd e p e n d so nk n o w l e d g e o f t h e y i ' s a n d t h ef^sa tt h e M t h ,( M - l )s t ,( M - 2 ) n d , e t c .s t e p s .A m e t h o d i sc a l l e da l i m - s t e p m e t h o d i f o n l y m - s t e p so f p r e v i o u s i n f o r m a t i o n a r er e q u i r e d .I n t h ef o l l o w i n g d i s c u s s i o n o n l y f o u r - s t e p m e t h o d s w i l l b ec o n s i d e r e ds i n c e m e t h o d s w i t hf e w e rb a c k v a l u e s c a nb e e x t r a c t e d f r o mt h ed i s - c u s s i o n b e l o w .

T h e m u l t i s t e p m e t h o d w i t h w h i c h w ea r ec o n c e r n e d h e r e i s k n o w na s t h ep r e d i c t o r - c o r r e c t o r m e t h o d .T h i s m e t h o d r e q u i r e sa f o r m u l af o rf i n d i n g af i r s t e s t i m a t e o f e a c h y .( h e n c e , t h e p r e d i c t o r ) ; a n dt h e n e v a l u a t i n ge a c h f u n c t i o n f .w e s u b s t i t u t et h i s i n t oaf o r m u l aw h i c h w i l l a d j u s tt h ev a l u e o b t a i n e d f r o m t h e p r e d i c t o r , h e n c e t h ec o r r e c t o r .y as t a n d a r d m e t h o d w e m e a n o n e i n w h i c ht h e s a m e s t e p - s i z e i su s e d i n a l l t h ee q u a t i o n s i n i n t e g r a t i n g f o re a c h y . .

F i r s to f a l l , r e w r i t et h es y s t e m o f d i f f e r e n t i a l e q u a t i o n s i n( 1 ) a sy . ' f^x, yry 2,. . . , y N) 1 7 )

w i t ht h e i n i t i a l c o n d i t i o n syi(xo }* yi(0) 1 8 )

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Wedesirethevaluesy(xh),i1 ,2 ,. . .Nwhereh i s again theincrementi n theindependentvariable.

Nextletus clarifysomenotationwhich wewillbeusing.

y.(M)- y .( xo*Mh)P

i1 ,2 ,. . . ,( M * l ) y.(xQ (M+l)h)-1 ,2 ,. . . , ( M ) y.'(xo+Mh)*f\(xo*Mh,y.(M),y2(M),. . . .N(M))' ( M + l ) 7.(xo*(M+l)h)f.(xo*(M*l)h,^ ( M + l ) ,. . , PNM*1)

( 1 9 )

( M + l )= yi(xo*(M+l )h) I 1 ,2 ,. . .N

B y using( 1 7 ) ,thedefiningequationsfor thestandardpredictor-correctoralgorithmaregivenby

P l) a iy.(M) b^M-l) Ciy. (M-2) d OMMiIe OO f^.^M-l) 20 )*xXi'01-2) Kiyi(M-3)], i ,2, ...,

C . (M* 1 ) 2y.( ) b2y.(M-l) C2y.(M-2)*[d2y.(M+l) e2y.(M) f 'QM) 21)*2yi,(M-2)], i1, 2, ...,

where( 2 0 )i sthepredictor,( 2 1 )the corrector,and th e coefficientsare to bedetermined.

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Using th estandardt e c h n i q u eof[10]forsolutionof thecon-st antcoefficients,th esolutionsetis :

a . 9-d .-3e. 3K. a29-1 5 d2-3e2bx9-9d. 2 4 KX b29-24 d2cx--1 7 9dx 3 e2-27^ c0-17 39d~ 3e.di-4 d2"d2 (22)el'el e2se2f:-18+dj+4 e2-1 7KX;f2-18 39 d2+ 4 egj*-6+Mj ex-1 4 KX;g2=-6+1 4 d2+e2Kl-Kl

Sincethecoefficientsin(22)areintermsoftheparameters d.,e.,Kj ,d2,e2,theconstantscanbedeterminedso astogivedesirablestabilitycharacteristics.

Thetruncationerrorinthepredictor-correctorequationsisgivenby

E P*TO9 di i 0K i>$is)Ec. 9-24d2-)h5y&

(23)

where cndErethepredictoran dthecorrectortruncationerror.p c * respectively.owever,thepredictoran dthecorrectorwerechosenindependently offourthorder.onsequently,fromHenrici[11 p261],thetruncationerrorofthealgorithmisgivenby hlone.

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1

B u t ,yl' maynot beavailable;thus,somemeansforreflecting thetruncationerror must be madeavailable.Thestandard techniquefo radjustingP.(M+1)withrespect to

thelocaltruncation errori st o add

E(M+1)* g PP . ( M )- C.(M))c ( 2 4 )

to P.(M+1),to improve its value.oticethefactthatE(M+1)will(S )not involve ySimilarly,the common methodofkeeping track of the totaltrun-cationerrori s to use

ET( M+1 )=g_p.(M*l)- C .(M+l))c p ( 2 5 )asan estimateof this errorat the( M + l )s t step,and from( 2 5 )criteriafo rhalving and doublingthestep-sizeh canbe generated.

According t o Crane andKlopfenstein[ 4 ] ,thevaluesforthe con-stantsi n( 2 2 )which yieldastabilitysituation notunlikethat ofRunge-Kuttamethods aregivenby

c ,

25 1 . 5 4 7 6 5 2 00 a2*1.0000S - 1 . 8 6 7 5 0 3 00 b2*0 . s 2 . 0 1 7 2 0 4 00 c2*0 . -0.6973 5 3000 d20. 3 7 5 000000a 2 . 002 2 4 7 00 620. 7 9 1 6 6 6 6 6 7

-2 . 0 3 1 6 9 000 f2* - 0 . 2 0 8 3 3 3 3 3 3s s 1 . 8 1 8 6 0 9 00 *2 =0. 0 4 1 6 6 6 6 6 6

.* -0.71 4 320000

e ,*

( 2 6 )

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wherethetrailingzerosareused to makethe methodas"numerically"fourthorderaspossible.hecoefficientsin thecorrectorequationdeterminethe standard Adams- Mculton corrector.TheGeneralizedPredictor-Corrector

By usingthe standardpredictor-correctorformulas asdiscussedinthe precedingsection,oneinherentshortcoming arises;thatofusingthe samefixedstep-sizehinintegration ofthevarious y . 's.In manyphysicalsituations,andparticularly in the mathematicalmodelrepresentingthetrajectory of aballistic rocket,the systemofdifferentialequationsthatarise mayhavesomesolutionsvary-ingmuchmorerapidly thanotherswithrespecttotheindependentvariable.incethestep-sizemustbechosento givethedesiredaccuracyi n the mostrapidlyvaryingsolution,byusingthe standardpredictor-corrector,computertimemaybewastedin integratingtheslowervaryingsolutions thruuseoftheshortstep-size h .hus,theneedarisesforageneralizedpredictor-correctoralgorithm inwhichthe step-sizesmaybedifferentineachequation.othisend,wewishtouseanincrement h .i n theithequationwhere h . m.h.,l-l li *i=2 ,3 ,. . . ,Nandm .i s apositiveinteger.Onewouldlike toprogress tothepoint xh .bymeansofonepredictor-correctorstepinthefirstequation,m-stepsi n thesec-ond equation,nu m-stepsinthethirdequation,. . , num_. . .m . .stepsi n theNthequation.hati s ,wedon'twish t o calculateeachf . y !at anypointsbetweenxM h .and x(M+l)h.,i1 ,2 ,. . . ,N .

Supposefo rthe momentwehave a methodofcalculatinghiy ^ r . )fo r r . h .*jjyj1 ,2 ,. . . ,j j - ,i1 ,2 ,. . . ,N-l.N

Letthismethod bedenotedby( * ) .e call( * )the generalized pre-dictoras willbe borneouti n thefollowingdiscussion.

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Toprogressfromxoxheuse thefollowingprocedure:hNput M 1 ,r . j j - ,i -1 ,2 ,. . . ,N- li

1 .redictyN( M)usingstandardpredictor2 .redict yi(ri)using( * ) ,i*1 ,2 ,. . . ,N-l3 .omputef N(M )4 .orrect yN(M)usingstandardcorrector5 .omputefN(M )using correctedyN(M )

putM = IM,r .=2 r .ti Nl,2 ,. . . ,N-ln u . timesN

6 .ompute * " Ni ( l )using(A )N( mN)( b )redicted yNi ( l )from standardpredictor( c ).(mN r . )from( * ) ,i1 ,2 ,. . . ,N-2

7 .omputef N. ( 1 )using corrected yN. ( 1 )

Thesameschemei srepeated t o advancefrom xh . ,,t o x2h k ,,N- l o N-l(using( * )advancedonestepfo r y Ni ) .ontinuinginthisfashionthenfindsthesolutionat xB L , ,h x . xh . we arenowo N - l N-l o N-2able t ocompute a correctedvaluefo r yN2atthis point.henwe

h .finallyreachxh . ,we haveevaluatedf .atonlypoints,

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i *1 ,2 ,. . . ,N .ora simpleexampleillustrating theabovepro-cedure seeAppendixA .Now i t onlyremains t o determinethegeneralizedpredictor( * )forcalculatingthepredictedvalueofy.(y),wherey i stheinter-

mediatevalueof M ,0< y

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e1( Y )- ej(Y)F^Y)=-M y) 4ej(Y)-1 7KM y4 5Y3 8Y2 4Y)g Y)* MX( Y ) ex(Y)-HKjM-1/2( y 4 + 4Y3 SY2 2 Y)

Thecorresponding truncationerrorinthegeneralized predictorisgiven b y :

E =12dx(Y)- 4ej(Y)+4 0 ^( 7 )5 4 3 2 h5y (S) ( y * 6Y4*iyA 12Y + 4Y)J 3 0 )

Asbefore,the standardtechnique for improving P.(Y)is toadd

BW ,-a,.(p... ) 31)toP .(Y) ,whereE .i sEty =1 ,Esfrom( 2 3 ) , .i s thepre-dietedvalue ofthepreviousstepandC .i s thecorrectedvalueofthepreviousstep.

tSimilarly, the truncation errorin standardcorrectori s reflectedi n

E T M + l ) . - J - L - ( , 0M) - C N( M + 1 ) )c c p 3 2 )whereEnd Ere the respective truncationerrorsin the standardpredictor-corrector.ycalculating E TM + l ) ,the validityofthecanswerfoundbyusingthe standardpredictor-corrector maybe checked.

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However,the taskof accountingfo rthetruncation errorinthestepsusingthegeneralized predictor i s somewhatdifferent.inceonly predictedvalues fromthegeneralizedpredictorareavailableat intermediate steps,the truncationerrori n thegeneralizedpre-dictor i s reflected in

* T M * FT # 4 * *0> i " l> 2 > " N - 13 3 )P f c c " f i l 1 x where E .is Ety 1 ,and P .and C .are thepredictedandcorrectedvalues,respectively,obtainedat thecurrentstep.hisvalue mustbeusedthroughoutthecurrentstage.

AnalysisofPredictor-CorrectorMethods

Onebasicshortcomingi n allpredictor-correctormethodsi sthatthey requiresome other procedure to obtainenoughvaluess o asthemethodcanbestarted.n thecaseofthegeneralized predictor-corrector,someproceduremust beusedtogeneratethe first3(m.)stepsinthe ithequation.ne methodofstartingtheproceudrei stheRunge-Kutta method.

Onedistinctdisadvantage ofthe predictor-correctormethodsi sthedifficulty inhalving the step-size h .y utilizing thegen-eralized predictor-corrector some ofthesedifficultiescanbe sur-mounted.till,sometimes halvingone oftheincrementsi s required,necessitatinginterpolationor evenrestarting thecurrentstep.Stability ofthepredictor-corrector algorithmswasnotdiscus-se dhere due to thelengths thatonei s required to go t o anade-quatediscussion.uffice i t tos a y ,the systemof differentialequationstobesolvedmustbecarefullyexaminedtodeterminewhatodd characteristicsexist and then thepropertypeof predictor-cor-rectoralgorithmchosento givethedesired stability andaccuracy.

For a furtherdiscussion see[ 1 0 ] .Predictor-correctormethodsin g e n e r a l , aremuchfasterthantheRunge-Kuttamethods andi n particular,use ofthe generalized predic-torwillreduce the functionevaluationsper stepand hencefurtherreducecomputer timeused.

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V

%\\

A

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Block m e t ho d sc a n also bea p p l i e dtop r e d i c t o r -c o r r e c t o rs c h e m e stoimprovetheir a c c u r a c y a ndefficiency.oram o r ecomplete dis-cussion of block methods,see[17],Hybrid Method

Hybrid methodsdifferfromt h o s e p r e v i o us l y d es c r i b e dinthatin additionto usingpreviousinformationfromth eM,(M-l)st,(M-2)nd,etc.steps,anoutsidemethodisusedtocalculateinformationatsome(M-y)thstep,0< y

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efficient useof thesemethods.lthougha smaller step-size i s gen-erally required for predictor-correctormethodsras compared to theRunge-Kuttamethod,theuseofthegeneralizedpredictor-con ectormethodmayalleviate many problems foundi n multistepmethods andmayusetheleast computer time ofallthemethods described.

The fairly attractivemethodsbrieflydescribed inthe finalsectionpoint outothermethods^ thatstemfromthe Runge-Kuttaandthemultistepmethods,that maybe investigated foruse ina ballisticrockettrajectory programduetotheircomposite ofthedesirablecharacteristicsof theRunge-Kuttrandmultistep methods.Atthepresenttimethe Atmospheric SciencesOfficeat White

Sands Missile Rangei s programming thegeneralized predictor-correc-toralgorithm forusein the six-degreeoffreedomballisticrocketmodel.ti sfeltthatthisalgorithm canbestaccomplishthedesiredtaskwith aminimum ofcomputertimendwith accuracycomparabletothatofRunge-Kuttamethodssedcurrently.

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14.ordsieck,Arnold,"OnN u m er i c a lIntegrationofO r d i n ar y Differen-tialE q u a t i o ns . "Math.Comp . .Vol.16,1962,pp.22-49.

15.ichardson,L .E.,a nd J.A.Gaunt,"TheDeferred A p pr o achtotheLimit." Trans,f t p v .Soc.London .Vol,226A,1927,p.300.16.omanelli,MichaelJ.,"Runge-Kutta M et ho d sfor theSolution of

O r d i n ar yDifferentialEquations,"Chapt.9ofM a t he m a t i c a lM e t h o d sforDigitalComputers ,Jo h n - W i l e yandSons,Inc.,New York,1 9 6 5 "

17.osser,J.Barkley," A Ru n g e - Ku t t aforallFisons,"MRCTechnicalS u m m a r yReportNo.698,September1966

18.alters,R a n d a l lK. ,"A G e n e r al ize dPr e d i ct o r - Co r r e ct o rMe t h o dfortheSolutionofO r d i n ar yDifferentialE q u at i o n s ,"tobep u b l i s h e dinT r an s act i o n softh eTwelfth Conferenceof ArmyM a t he m a t i c i a ns .

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APPENDIX AAn E x a m p l e U s i n g t h e GeneralizedPredictor-Corrector Method

S u p p o s e w e h a v e t h e f o l l o w i n g s y s t e m off i r s t orderd i f f e r e n t i a l e q u a t i o n s

Vfix'x>2'3}y 2*f 2( x ,yvvy 3 4 )yZ *f3(x'yV2 > y3 }

w h e r e y 2 v a r i e s t h r e e t i m e s a s rapidlya s y .withr e s p e c t t o x ,a n dy - v a r i e s t w i c e a s rapidlya s y 2withr e s p e c t t o x .e w o u l d l i k et o f i n d t h e s o l u t i o n of( 3 4 )a t t h e p o i n t xh . .h e r e f o r e ,l e th .= 3 h2a n d h2 = 2 h _ whereh .i s t h e i n c r e m e n ti nt h e i t h e q u a t i o n .T h i s means t h a t U= 3and m . =2 .h e f o l l o w i n g f i g u r e w i l li l l u s - t r a t e t h i s s i t u a t i o n .

xr t +hio 1h ' >

y2;

y-if

x xh ~ x2 h ~ x3 hno o 2 o 2 o 2x xh , 6 h >o o 3 3

I nf i n d i n g t h e s o l u t i o nof( 3 4 )a t xh .w e w i l l c a l c u l a t e f .a t xh .o n l y ,La t xh - a n d x2 h ~ o n l y a n d f .a t xK h7,O 1 ' * 0 3K =1 ,2 ,. . . ,6 ,a n d u s e t h eg e n e r a l i z e dpredictor f o r f i n d i n g t h ey ' s betweent h e s ep o i n t s .

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T h e d e t a i l e d procedure i s t h e f o l l o w i n g :h31 .u t r .*r ,i 1 ,2i

2 ..r e d i c t y1 ) usingt h e s t a n d a r d predictorb .r e d i c t y x(r^=y x( 1 / 6 ) using( * )c .r e d i c t y 2( r2)*y 2( 1 / 2 ) using( * )

3 .o m p u t e f _( 1 ) ,c o r r e c ty -( 1 )using t h e s t a n d a r d correctora n d t h e n c o m p u t ea f i n a l v a l u e off .( 1 )4 .u t r .= 2 r .=m - r . ,i * 1 ,21 i ox5 ..r e d i c t y -( 2 )*y _( m . )u s i n g t h e s t a n d a r dpredictor

b .r e d i c t y J( r j )=yx ( 2 / 6 )using( * )c .r e d i c t y2(rj=y 2( 1 )u s i n g t h e standardpredictor

6 .o m p u t e f -( 2 ) ,c o r r e c t y -( 2 )usingt h e s t a n d a r d c o r r e c t o r ,a n d t h e n c o m p u t e af i n a l valueoff _( 2 )

7 .o m p u t e f2( 1 )u s i n g 5 ,c o r r e c t y 2( 1 )usingt h e s t a n d a r dc o r r e c t o r ,a n d t h e n c o m p u t e af i n a l valueo f f2( 1 )

I nt h e s e c o n d e q u a t i o nw e a r e n o w a t t h e p o i n t xh .e nowa d - v a n c e( * )o n e s t e p w h e n u s i n g i ti nt h e s e c o n d e q u a t i o na n di tw i l lb e understoodt h a t w h e n w e s a y " p r e d i c t y 2( r2) ,0 < r 2< l ,using( * )Mh ' e meanw e a r e p r e d i c t i n g t h ev a l u e o f y ~ betweenxh0a n d x2 h22 o 2

h3 '8 .u t rx =ZT1 =( m3 1 )tyr 2=^-9 ..r e d i c t y,. ( 3 )u s i n g t h e s t a n d a r d predictor

b .r e d i c t yx (r =yx ( 3 / 6 )u s i n g( * )c .r e d i c t y2( r2)=y2( 1 / 2 )u s i n g( * )

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1 0 .ompute f -( 3 ) ,correct y-( 3 )using the standardcorrectorandthen computeafinal valueof f .( 3 )

1 1 .ut r .*4rx=2m3r.,2 2r2 m7r21 2 ..redicty .( 4 ) y _( 2 m . )usingthestandardpredictor

b .redictyx(r^=yx( 4 / 6 )using( * )c .redicty2( r 2) y 2( 1 )usingthestandardpredictor

1 3 .ompute f ,( 4 ) ,correcty-( 4 )usingthestandardcorrector,andthencomputeafinalvalueoff_( 4 )

1 4 .omputef1 )using1 2 ,correcty2( 1 )usingthestandardcorrector andthencomputeafinalvalueoff ~( 1 )

Again weadvance( * )onestepwhenusingi tinthesecondequation.Wewillnow computey2( r2) ,o< r2r2 -m

1 6 ..redict y -( 5 )usingthestandard predictorb .redictyxr =yx ( 5 / 6 )using( * )c .redicty2( r 2)*y^ ( 1 / 2 )using( * )

1 7 .omputef-( 5 ) ,correcty( 5 )usingthestandard corrector,andthencomputeafinalvalueof f -( 5 )

1 8 .ut r . 6 r ,*m-m2 r . ,r2=2r2=m-r21 9 ..redicty -( 6 )=y-( m - n u ) usingthestandardpredictor

b .redict y .( r . )=y .( 1 )usingthestandard predictorc .redicty2( r 2)=y2( 1 )usingthestandardpredictor

2 0 .rom 1 9 ,computef .( 1 ) ,correct y .( 1 )usingthestandardcorrector,andthencomputeafinalvalueoff .( 1 ) ,i=1,2,3.

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Wehavereachedour solutionatxhymeansof one standard pre-dictor-corrector stepi n thefirstequation,m2 *3 standard predictor-corrector stepsi n the secondequation,andm2m_=6 standardpre-dictor-corrector stepsinthethirdequation.

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137.ema,J.,"A nAcousticRay TracingMethodorigitalomputation,"eptember 1965 .138.ebb,W .L .,"Morphologyo fNocti lucentClouds," /.Geophys .Res . ,0,8,463-4475 ,eptember965.139.ays,M .,andR .A.Craig,"O nth eOrdero fMagnitudeo fLarge-ScaleVerticalMo-t ionsnheUppertratosphere,"J.Geophys.Res. ,0,8,4453-4462,September965 .140.ider,L .,"Low-LevelJetatWhiteSandsMissileRange,"September1965 .141 .amberth,R .L.,R.Reynolds,andMor ton urtele,The ountaineeWavetWhiteandsMissileRange,"Bull.Amer.MeteoroiSoc.t46 ,0,Octo-ber965.142.eynolds,R .ndR . . amberth,AmbientTemperatureMeasurementsfromRa- diosondes lownnConstant-LevelBalloons,"October1965 .143.cCluney,.,Theoreticalrajectoryerformancefhe ive-InchGun robeSystem,"ctober965.144 .ena,R .ndM.Diamond,"Atmosphericound ropagationnearth eEarth'sur-face,"October1965 .145 .ason,J.B .,"AStudyo fth eFeasibilityfUsingRadarChaffo rtratosphericTemperature easurements,"November965 .146 .iamond,M .,an dR . . ee ,Long-Rangetmosphericound Propagation,".Geophys.Res. ,0,2,November 1965 .147.amberth,R . .,"Onth eMeasuremento fDustDevilParameters,"November1965 .148 .ansen, .V.,nd . .Hansen,Formationo fanInternalBoundaryoverHeter-ogeneousTerrain,"November1965 .149.ebb,W.L .,"Mechanicso fStratosphericSeasonalReversals,"November1965 .150...ArmyElectronicsR&DActivity,U.S.ArmyParticipationinth eMeteoro-logicalRocketNetwork,"anuary1966 .151 .ider,L.J.,andM.Armendariz,"Low-LevelJetWindsat GreenRiver,Utah,"Feb-ruary1966 .152.ebb,W . .,DiurnalVariationsnheStratosphericCirculation,"February1966 .153.eyers,N.J.,B .T .Miers,andR .J.Reed,DiurnalTidalMot ionsnearth eStrato-pauseDuring48HoursatWSMR,"February1966 .154 .ebb,W.L .,"TheStratosphericTidalJet,"February1966 .155 .all,J.T,"FocalPropertieso faPlaneGratingn Convergent eam,"ebruary1966 .156 .uncan,L.D.,an dHenryRachele,"Real-TimeMeteorologicalystemorFiringo fUnguidedockets,"ebruary966 .157.ays,M.D.,"ANoteonth eComparison o fRocketandEstimated GeostrophicWindsatth e10-mbLevel,"J. Appl.Meteor . ,February1966 .158 .ider,L., andM.Armendariz," AComparisonf ibalndTowerWindMeasure-ments,".ppl.Meteor . ,,February1966 .159 .uncan,L .D.,CoordinateTransformationsn rajectoryimulations,"ebruary1966 .160.illiamson, .E .,Gun-LaunchedVertical robestW hiteandsMissileRange,"February966.16 1 .andhawa,.,,zoneMeasurementsithocket-Bornezonesondes," arch1966 .162.rmendariz,Manuel ,and aurenceJ.Rider,"WindShearfo rSmallThicknessLay-ers,"March966.163.ow,R .D.H .,"ContinuousDeterminationfheAverageoundVelocityvernArbitraryath,"March966 .164 .ansen,rankV.,RichardsonNumber ableso rheurface oundary ayer," March966.165 .ochran,V.C,E.M.D'Arcy,andFlorencioamirez,Digitalomputerrogram fo r Five-Degree-of-Freedom rajectory,"March966 .166 .hiele,O.W .,andN.J.Beyers,"Comparisono fRocketsondeandRadiosondeTemp-eraturesandaVerificationo fComputedRocketsondePressureandDen-sity,"April966 .167.hiele,0.W .,ObservedDiumalOscillationso fPressureandDensityinheUpper StratosphereandLowerMesosphere,"April966.168 .ays,M.D.,andR.A.Craig,"OntheOrderfMagnitudefarge-ScaleVerticalMotionsintheUpperStratosphere," J.Geophy.Res. ,April1966 .169 .ansen,F.V .,"TheRichardsonNumbernhe lanetary oundary ayer,"M ay 1966 .

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170.allard,H .N.,"TheMeasuremento fTemperaturenhetratospherend eso-sphere,"June1966 .171.ansen,FrankV.,"TheRatioo ftheExchangeCoefficientsfo rHeatandMomentuminaHomogeneous ,ThermallyStratifiedAtmosphere."June966 .172.ansen,FrankV.,"Comparisono fNine rofileModelsorheDiabatic oundaryLayer,"June1966 .173.achele,Henry,Aound-Ranging echniqueorocatingupersonic issiles,"May966.174.arone,W.A .,andC.W.Querfeld."ElectromagneticScatteringfromInhomogeneousInfiniteCylindersatObliquencidence,"J.Opt .Soc.Amer.6 ,4 ,476-480,April1966 .175.ireles,Ramon,"Determinationo fParametersnAbsorptionpectrayNumericalMinimization Techniques,"J. Opt .Soc.Amer.56 ,5, 644-647,May1966 .176.eynolds,R .,ndR. . amberth,AmbientTemperatureMeasurementsfromRa- diosondes lownnConstant-Level alloons,"J. ppl. feteoroL,5,3, 304-307,June966 .177.all,JamesT .,"FocalPropertieso faPlaneGratinginaConvergentBeam,"Appl. Opt . ,5,051 ,June1966

178.ider,LaurenceJ.,"Low-LevelJetatWhiteandsMissileRange,"J. ppl.Mete-orol.,5,3,283-287,June966 .179.cCluney,Eugene,"ProjectileDispersionsCausedbyBarrelDisplacementinth e5-InchGunProbeSystem,"July966 .180.rmendariz,Manuel ,an dLaurenceJ .Rider,WindhearalculationsormallShearLayers,"June1966 .181 .amberth,R oy .,andManuelArmendariz,UpperWindCorrelationsnheCen-tralRockyMountains,"une966.182.ansen,FrankV .,andVirgilD.Lang,TheWindRegimeinth eFirst62Meterso fth eAtmosphere,"June966 .183.andhawa,JagirS .,"Rocket-BorneOzonesonde,"July966 .184 .achele,Henry,andL.D.Duncan,"TheDesirabilityo fUsinga as tamplingRatefo rComputing WindVelocityfromPilot-BalloonData,"July1366 .185 .inds, .D.,andR .G .appas,"AComparisonfhree ethodsorheor-rectionfadarlevationAngleRefraction rrors,"August966 .186 .iedmuller,G . .,ndT . . arber,AMineral ransitionnAtmosphericDustTransport."August966.187.all, J.T ., C.W. Querfeld, and G.B.oidale,Spectralransmissivityfhe Earth'stmospherenhe50o00WaveNumbernterval," ar tIVFinal),July966 .188 .uncan, .D.nd . . ngebos,TechniquesorComputingauncherettingsfo rUnguidedRockets,"eptember966.189 .uncan, .D.,BasicConsiderationsnheDevelopmentfnnguidedRocketTrajectory SimulationModel,"eptember966.190.iller,WalterB .,Considerationf^me roblemsnCurve itting,"eptember 1966 .191 .ermak,J.E.,andJ. D.Horn ,"TheTowerShadow Effect,'August1966 .192.ebb,W.L .,StratosphericCirculationResponseto aSolarEclipse,"October1966 .193.ennedy, ruce,MuzzleVelocityMeasurement,"ctober966 .194 .raylor,LarryE.,"ARefinementTechnique for Unguided Rocket Drag Coeffic-

ients,"ctober966 195.usbaum,Henry,AReagentorheimultaneousMicroscopeDeterminationfQuartzndHalides,"October966 .196 .ays,MarvinndR.O.Olsen,Improved Rocketsonde Parachute-derivedWindProfiles,"October966.197.ngebos, ernard.ndDuncan,ouis.,Aomogramor ieldDetermina-tionfLauncherAnglesorUnguidedRockets ,"October966.198 .ebb,W. .,MidlatitudeCloudsintheUpperAtmosphere,"November966 .199 .ansen, rankV.,The ateralntensityo fTurbulenceas aFunctiono fStability,"November966.200.ider, ..ndM.Armendariz,Differencesf owernd ibalWind rofiles,"November966.201.ee,Rober t .,AComparisonfEight Mathematical Models for Atmospheric Acousticalay racing,"November966.202.ow,R .D.H .,tl. ,AcousticalndMeteorologicalDataReportSOTRAN andII,"November1966 .

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203. Hunt,.A.ndJ.D.Horn ,Drag iate alance,"ecember966.204rmendariz,M .,andH.Rachele,Determinationf RepresentativeWind rofilefrom alloonData,"December966 .205ansen, rankV .,"TheAerodynamicRoughnessfheComplex errainfWhiteSands issileange,"anuary967.206orris,ames .,WindMeasurementsnheubpolarMesopauseRegion,"an -uary1967.207al l James.,Attenuationf illimeterWavelength Radiation by GaseousWater,"anuary967.20 8hiele,0.W .,an dN. .Beyers,UpperAtmosphere ressureMeasurementsWithThermalConductivityGauges,"January967.209rmendariz.M .,ndH .Rachele,DeterminationfaRepresentativeWindProfilefrom alloonata,"anuary1967.21 0ansen, .V .,TheAerodynamicRoughnesso ftheComplexTerraino fWhiteSandsMissileRange,NewMexico,"January1967.21 1'Arcy EdwardM .,SomeApplicationso fWindoUnguidedRocketImpactPre-diction," arch967.21 2ennedy*Bruce,"OperationManualfo rtratosphereemperatureonde," arch1967.21 3oidale,G .B .,S.M.Smith,A.J.Blanco,andT .L. arber,"AStudyo fAtmosphe-ricDust,"March967.21 4ongyear,J.Q. ,AnAlgorithmfo rObtainingolutionsoaplace'sitadqua-tions,"March967.21 5ider, .J. ,AComparisonf ibaiwithRaoban dRawinWindMeasurements," April967.216 .reeland,A.H .,ndR.S. onner.Resultso fTestsInvolvingHemisphericalWindScreensnheeductiono fWindNoise," April1967.217.ebb,Willis .,ndMaxC. olen,The-regionair-W eatherlectricield," April967.218.ubinski, tanley .,AComparativeEvaluationo fth eAutomaticTrackingPilot-BalloonWindMeasuringystem,"April1967.219.iller,WalterB .,andHenryRachele,"O nNonparametricTestingo fth eNatureo fCertainTimeSeries,"April967.220.ansen, rankV.,SpacialndTemporalDistributionfth eGradientRichardson NumberinheSurfaceandPlanetaryLayers,"May1967.221.andhawa,agirS .,DiurnalVariationfOzoneatHighAltitudes,"May967.222.allard,arold. ,AeviewfevenapersoncerningheMeasurementfTemperaturenhetratosphereandMesosphere,"May967.223.illiams, enH .,SynopticAnalyseso fheUppertratosphericCirculationDur-inghe at eWintertorm Periodo f1966,"May1967.224.orn,J.D.,andJ.A.Hunt,"SystemDesignorhetmosphericciencesfficeWindResearch acility,"May967.225.iller,Walter .,ndHenryRachele,DynamicEvaluationfRadarndhoto Trackingystems, May967.

226.onner,Robert .,ndRalphH.Rohwer ,"AcousticalandMeteorologicalDataRe- por t OTRANIIndV,"May967.227.ider,L.J.,"OnTimeVariabilityo fWindatWhiteSandsMissileRange,New Mex-ico,"une967.228.andhawa,JagirS.,MesosphericOzone easurementsuring olarclipse,"June1967.229.eyers,N.J.,ndB .T .Miers,ATidalExperimentinheEquatorialStratosphereoverAscensionsland8S)" ,June1967.230.iller,W. .,andH.Rachele,"O nth eBehavior o fDerivativeProcesses,"June1967231.alters,RandallK.,NumericalntegrationMethodsor allisticRocket rajec-tory imulation rograms,"une967.

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