THE NUMERICAL SOLUTION OF ORDINARY AND ALGEBRAIC .
D U B L IN C IT Y U N IV E R S IT Y , D U B L INSchool o f M athem atical SciencesM . Sc. TH ESISTHE NUMERICAL SOLUTION OFORDINARY AND ALGEBRAICDIFFERENTIAL EQUATIONS USINGONE STEP METHODSbyGerard K eogh B . Sc.Supervisor: D r. John Carroll, School o fM ath em atical SciencesThis Thesis is based on the candidates own workSeptember 1990
A ck n ow led gem en ts.I w ould like to acknow ledge and th an k the following for their help and assistancewhile doing this workM y su pervisor D rJo h n C arro llT h e staff at D C U , esp ecially Pau lin e O ’ G o rm anT h e p o stgra d u a te students of m ath em atics at D C UT h e staff of th e M ath em atics D eptat C o rk R T CM y parents and fam ily, esp ecially m y sister M aureenF in a lly, Siobh an, for her kindness, love and care over m a n y m onths, a v e ry specialth an k you m y darling
A b stractT h is thesis addresses th e problem of finding num erical solutions to ord in ary andalgeb raic differential equation system sO u r p rim a ry focus is the application o f one-step num erical schemes to these problem classesF ir s tly w e concentrate on the narrower class of explicit O rd in ary D ifferential E q u a tion (O D E ) system sW e an alyse the th eory necessary develop efficient algorithm sbased on our chosen one-step num erical schemesT h e se algorithm s are then appliedto th e solution o f a stan d ard test set o f O D E system sT h e results are then com paredw ith those obtained using stan d ard softw are packages on th e sam e problem test setO ur th eory is then extended to include th e w ider class of A lg e b ra ic D ifferentialE q u a tio n (m ore com m only called D ifferential A lg e b ra ic E q u a tio n ( D A E ) ) system sB ased on this theory, w e are able to ad ap t our one-step schemes to solve this harderclass o f problemO nce again the resulting algorithm s are tested on a selection ofproblem s and results are com pared w ith those o btained fro m stan d ard softw are p ack agesO n all problem s considered, we d em onstrate th a t our techniques can oftenprovide efficient alternatives to the m ore com plex m ethods adopted in the standardsoftw are packages designed for these problem classes
C on ten ts1 In trod u ction .231 1Systems of differential e q u a t i o n s .1 2O b je ctive s an d review. . .3T h e num erical solu tion o f O rdinary D ifferential E quations (O D E s).52 1T h e th eory o f O D E s52 2E x iste n ce an d uniqueness.2 3D iscrete variable m e t h o d s .2 4O rder an d convergence2 5S ta b ility o f num erical m ethods. . . . .6.7.9.2 5 1S ta b ility properties o f th e linear test equation2 5 2S -sta b ility. . . .11.11.2 6Im plem entation o f num erical m ethods2 7E rro r m easurem ent and Stepsize Strategies13.15.18V ariable step integrators for th e solution o f stiff O D Es.203 1Introduction20.3 2T h e S tro n g ly S -sta b le D I R K ( 2 ,2 ) schem e3 3T h e C om p osite Integration 0 - B D F 2 schem e.3 4E rro r estim ation3 53 63 7.20.E rro r estim ate for the D I R K ( 2 ,2 ) schem e3 4 2E r ro r estim ate for th e C o m p o site Integration schem eSo lvin g the nonlinear equations.24T h e nonlinear equations arising from th e D I R K ( 2 ,2 ) schem e3 5 2S o lvin g th e nonlinear equations o f th e C om p o site Integration3 5 3O th er asp ects o f solving the nonlinear system s.A lg o rith m D I R K ( 2 ,2 )3 6 2A lg o rith m C om p o site Integration schem e3 6 3E stim a tin g the initial steplengthN u m erical experim ents.2626.Introduction4 2Infinitely S tiff O D E s.4 3L in ear C o n stan t Coefficient D A E s4 4L in ear non constant coefficient D A E system s283030f4 147. .1.D ifferential A lgebraic E quations (D A E s). .26.2425V ariab le step algorithm s for the solution o f O D E s .3 6 124243 5 1.22243 4 .1schem e4. . .22.474951.54
4 54 64 75.58In itial conditions for the linear constant coefficient problem584 6 2Initial values for the general problem60F in d in g th e in dex o f D A E system s. .61645 1IntroductionE rro rs in S o lvin g D A E s n u m e r i c a l l y .5 3E rro r E stim a te s for D A E s.5 4. . . . . .L in ear S ta b ility for D A E s. . . . .Im plem entation o f Im plicit Schem es for D A E s5 6T h e Iteration M a trix and Scalin g.6 5. .6870. . . .71.Initial conditions for N u m erical S c h e m e s .N um erical schem es for solving D A E s.Introduction. .6 2D I R K ( 2 ,2 ) schem e for D A E s6 3T h e C om p osite Integration Schem e for D A E s6 4ODEPACK.37577. . .7776 1& LSO D I64.5 56 6AInitial conditions for D A E system s4 6 15 26 5756N um erical A sp ects o f Solving D A E s.5 76T h e general Im plicit D ifferential E q u a tio n.77.8082D A SSL84D A E test problem s an d results6 6 1R e ca st Index - 1 problem s6 6 2O th er In d e x -1 problem s6 6 3In d ex-2 problem s.86.8692.96C onclusions and Future D irection s.7 .1Introduction7 2R e view and Conclusions101. .10 1. . . .10 17 3T ensor m ethods for solving N onlinear S yste m s . .7 4Exten sio n s of D A E problem s.E quivalence o f D IR K (2 ,2 ) and C om posite2.10 4.10 6Integration schem es. 108
C hapter 1In trod u ction .1.1System s o f differential equations.T h e tim e beh aviour o f m an y n atu ral and tech nical processes, can b e described b ysystem s o f O rd in ary Differential Eq u atio n s (O D E s )In general, tw o typ es o f system sarise. T h e explicit first order systemy '(0 f(* y(*)) w ith y ( a ) givenFo r this system , y ( i ) ,t\a,b] (11)y '(t) and f ( ) are n — dim en sio n a l vectorsT h e second system is th e general im plicit O D E given b yF ( i , y ( * ) , y '( * ) ) 0w ith b o th yt e [a, 6](1 2 )(a) and y '(a) given. O nce again y ( t) ,y '( t ) and F ( ) are n — d im en sio n a lvectorsT y p ic a lly explicit system s ( 1 1 ) and im plicit system s ( 1 .2 ) arise in sim ilar ar easF o r exam ple, electronic circuits can be m odelled b y system s o f O D E sD yn a m icelem ents such as capacitors an d inductors generate differential equations w hile the in clusion o f sta tic elements in the circuit give rise to algebraic equationsT h e algebraicequations are coupled to the differential equations form ing a D ifferential A lg e b ra icE q u atio n ( D A E ) system , see C am p b e ll [12]C on tro l problem s, solved b y variation al techniques, provide us w ith another e xam ple of ord in ary differential system sto explicit system sIn som e cases, the E u le r-L a g ra n g e equations leadH owever, th e best known control problem , is the linear q u ad raticregulator 1x' A x —B ux (a ) x aw ith the associated cost functionalJ (u ) jj x '/ f x u ujdtT h e m atrices H and Q are sym m etric and po sitive definiteU sing th e th eory ofL a g ra n g e m ultipliers, the necessary conditions for a m inim um , see C am p b e ll & M eyer1We drop the dependence of x and u on t for clarity3
[11], arex' Ax Bnx (a ) x aA' —At\ — H xA(b) 00 B lA Q uO nce again, w e obtain a system o f differential algeb raic eqautionsW h e n a system of tim e dependent P a rtia l D ifferential E q u a tio n s ( P D E s ) , aresolved using an ap p roxim ate technique, such as finite differences or finite elements[59], a system o f differential equations arisesIf the system is a coupled system o f el lip tic an d p arab olic P D E s , then the resulting equations generated b y the ap proxim atetechnique are differential algebraic, see Petzold & L o tste d t [59]T h e final exam p le w e introduce is the sin gu larly p ertu rb ed scalar differential sy s tem g(t,y,z, )tz' h(t,y,z, e)y'w ith bothtE[a,y(a) and z(a) given U sually, e is a sm all p aram eter w ith e C 1 Syste m so f this form prove to b e u nsuitable for num erical solutionthe p ertu rb atio n p aram eter,representable num berp o rtan t(13)b]T h e reason for this is th ate m ay take a value sm aller than the sm allest m achineT h e n the q u alitative assessm ent o f the solution becom es im T h e lim iting casee 0 m ust be u nderstood and the resulting D A E systemsolved num erically1.2O bjectives and review .In this thesis, we concern ourselves w ith the solution o f explicit O D E system s 2 ( 1 1)and D A E system s, w hich can be w ritten in the formE y ' i{t,y(t))w ith y ( a ) and y '( a ) givent [a,b]In general the m a trix E is sin gu lar(1 4 )T o this end, we willd raw on theoretical results for the a n alytic solution o f b oth ( 1 1 ) and ( 1 .4 ) , wherenecessaryO u r m ain o b jective is to use the th e o ry given in this w ork to developnum erical m ethods for the solution o f O D E and D A E system sW e evalu ate theperform ance o f the techniques we propose again st som e stan d ard algorithm s availablefor th e num erical solution of these problem sIn C h a p te r 2 , w e stu d y explicit O D E s an d their num erical solutiontra te on ’stiff’ O D E system sW e concen T h e se problem s are sim ilar to the sin gu larly pertu rb edsystem s, but th e y are suitable for num erical solutionC o n cep ts o f convergence, ordero f a c cu ra c y and sta b ility will be discussed for num erical m ethods applied to O D E sW e show how num erical m ethods can b e im plem ented to solve exp licit O D E system sC h a p te r 3 develops the one step num erical m eth ods th at form the core o f thisthesisA g a in , we an alyze the a c cu ra cy and sta b ility o f these schem esalgorithm s based on the one-step form ula proposedwell known problem s th at have appeared in the literatu re2We simply call these ODEs for the remainder of this thesis4W e give twoF in a lly, w e test th em on some
T h e em phasis changes in C h a p te r 4, where w e consider th eoretical aspects ofD AEsT w o im p o rtan t topics are addressed in C h a p te r 4.th e index, or degreeof co m p lexity o f a D A E and the difficulties associated w ith initial conditionsWeoutline a selection o f techniques th at have appeared in th e literatu re for dealing w iththese problem sO nce again in C h a p te r 5 we return to num erical m ethods. W e explain w h y someD A E s are solvable b y num erical m ethods suitab le for exp licit O D E s and others arenotW e show th at the index or degree o f co m p lexity o f a D A E , determ ines both thea c c u ra cy and stab ility o f a p articu lar num erical schem e.C h a p te r 6 parallels C h a p te r 3W e exten d our one step schem es to the solutiono f system s o f the form ( 1 .4 ) . A g a in , we stu d y th e a c cu ra cy o f these schem es, usingthe th eo ry developed in C h a p te r 5 . W e outline how we intend to change our one stepschem es so th at th e y can b e used to solve D A E sW e then consider tw o well knownalgorithm s, which are available as Fo rtran routines for th e num erical solution o f D A Esystem sT h e se algorithm s are called D A S S L (D ifferential A lg e b ra ic S y ste m Solver)an d L S O D I (Liverm ore Solver for Im plicit O D E s )F in a lly, a w id e selection o f testproblem s are proposed and solved using all algorithm s outlined in this thesisT h e last C h a p te r, discusses how successful w e feel our softw are has been m solvingth e problem s consideredW e discuss where we feel progress can b e m ade in th e fu tu rein solving D A E system s an d close w ith a discussion o f possible extensions o f D A Ety p e problem s which to our knowledge have not ap peared in the literature5
C hapter 2T h e num erical solu tion o fO rdinary D ifferential E qu ations(O D E s).2.1T he theory o f ODEs.In this C h a p te r it is our intention to exam ine the th eo ry o f O D E s along w ith somenum erical m ethods for their solutionIn p articu lar w e are concerned w ith solving thegeneral first order nonlinear vector O D E o f the form ,y '(t) f ( t , y ( i ) )(2 1 )w herey (t)R -»R nandf ( , y(i))R x R n - R n,su b je ct to th e conditionsy(a) y ttandtG[a,b].T h e first question we ask ourselves regarding (2 1 ) is, does a solution y ( t ) existan d, if so, is it uniqueIn section 2 we outline the conditions th a t w e need toim pose on (2 1 ) for a unique well-posed stable solution to existW e also exam ineth e concept o f stiffness which is ve ry im po rtan t for th e num erical solution o f O D E sSection 3 introduces discrete variable (num erical) m ethods for the solution o f ( 2 1 )In p articu lar w e introduce two well known classes of m ethods, th e R u n g e K u t t a (R K )m ethods and the L in ear M u ltistep M eth ods ( L M M )Section 4 discusses the error,order an d convergence o f num erical m ethods w hen applied to ( 2 1 )S ta b ility ofnum erical m ethods is introduced in section 5 . W e give several definitions o f sta b ilityand d em o n strate their usefulness through relevant exam plesSection 6 deals w iththe im plem entation o f num erical m ethods. F in a lly in Section 7 , we look a t som e welldocum ented techniques for estim atin g th e error in th e num erical in tegration of ( 2 1 )and the associated problem o f stepsize adjustm ent.6
2.2E xisten ce and uniqueness.W e assum e th at fconstant(t, y (t)) is Lip sh itz continuous on [a, b], th at is there exists aL such th atl f ( * ,y ( 0 ) - f (*»z (0)lloo L y(i) - z (t) oofor all i [a,(2 2 )b] an d all y (t), z (t) (E R n l.M o re specifically if w e require the first p artial derivatives o f f ( - )a constantK th a t is, Kfor allt6[a,b e bounded b yb] and all y(t)6l i,] n(2 3 )R n, then (2 2 ) an d (2 3 ) gu arantee a unique, wellposed (in the sense th at the solution can b e m ade as accu rate as possible b y keepingp ertu rb atio n s sm all) solution to problem (2 1 ), see G e a r [31]T h e m ost im portant a ttrib u te o f (2 1 ) we are concerned w ith isstiffnessW hensolving O D E s num erically, stiffness will d ictate how well a num erical m eth od willperform on the O D ET o determ ine w hether or not ( 2 1 ) is stiff, we need to knowsom ething abou t th e natu re o f its solutions m the neighbourhood o f a p articu larsolution y ( t )H all &; W a tt [38] consider such a neighbourhood w h ere (2 1 ) can beclosely ap p roxim ated b y the linearized variation al equationsy '( i ) - J ( i ) [ y ( i ) - y ] - f ( i , y ) 0(2 4)J ( t ) is the Ja co b ia n m a trix o f p artial derivatives, evalu ated at (t, y )Remark 2 1 W e deal o n ly w ith stiff problem s m this thesis N o n -stiff O D E s arew hereb etter solved b y num erical m ethods such as A d a m s form ulae, see [31]J(t) in an interval o f t is sufficiently sm all, the localized eigensolutions o f (2 4) are ap p roxim ately exponentials e A‘ 4 , w here the A [s are th e localizedIf the variation o feigenvalues o f the Ja c o b ia n m a trix, assum ed w ith out loss o f gen erality to b e distinctT h u s th e solution yo f (2 1 ) in a neighbourhood o f the exact solution y ( i )of the formatt areny y (0 E c,eA,tVj» iw here th e c,th atare constants and the v ,Re(A ,), 0V z are the eigenvectors o fJ(t)If we assum el(l)rz , then clearly the com ponents o f th e solution y will1 /R e(\t) , these are called the time constants ofthe system T h e O D E (2 1 ) is stiff , if we have w id ely differing local time constantsd ecay at different rates, given b y It is the range m th e local values o f the ’’ tim e co n stan ts” o f a problem th a t providesa m easure o f stiffnessDefinition 2 1 (Lambert [46]) T h e O D E ( 2 .1 ) is said to b e stiff in an interval Iof [a, b] if, for t E I , we haveRe(A,) 0i l(l)n h e Maximum norm is sufficient for the type of functions we consider, however the Supremumnorm may be more appropriate in certain situations7
and max.,1,„ fle(A ) Qm m , linR e ( A,)w here the A ,’s are th e eigenvalues o f the Ja co b ia n m a trix o f f ( ¿ , y ( í ) ) , evalu ated onS (t) is called the local stiffness ratio o f th e problem ,see L a m b e rt [46] P roblem s m a y b e m argin ally stiff if S(t ) is 0 ( 1 0 ) w hile stiffnessratios o f 0 ( 1 0 6) are not uncom m on in p ra ctica l problem s Som etim es a problemthe solutiony (t) at tT h e ratiow h ich is stiff is referred to as a problem w ith ’’ w id ely differing tim e co n stan ts” , or asa system w ith a ’’ large L ip sh itz con stan t” , sincedf'f) dyw herep( ) is the sp ectral radius of the Ja c o b ia n o f f (-)2.3D iscrete variable m ethod s.W ith o u t loss o f generality we consider the sca lar version o f (2 1 )y1 w itht e [a, 6 ](2 5)y(a) ya T h e e x a ct solution o f (2 5 ) is approxim ated on set o f discrete points(X 05 1) 25 * 5 /bIf th e discrete variable m ethod 2 approxim ates th e tru e solutiony (tn) at th e pointt n b y yn, then we shall consider the class o f discrete variable m ethods given b yyt s,(h ),and0 i k — 1 , sta rtin g valuesk 1 a t Vn t « 0h*f f (tn, y n ki '( 2 *6 )0 n N — k,whereh i n 1 — t n and N h tn — t0 If yn k does not ap p ear in / () 3 then ( 2 6 )is said to b e explicit otherw ise it is im plicit. T h e above class o f m ethods contains areasonably w ide selection o f the m ost popular discrete variable m ethods, (see H all &W a tt [38])For exam p le the general im plicit one-step m ethod, com m only known asth e B a c k w a rd E u ler ( B E ) m eth od, is defined b yy»» l —Vn "I" h f ( t n l: 2/n l)W e consider two im p o rtan t subclasses o f (2 6 ).K u t ta ( R K ) m ethods(2 7)T h e first o f these is th e R u n geT h e idea behind the R K m ethods is to in tegrate fromt n tot n i tn h, b y ap p roxim atin g the integral mf tn 1y( n i) y{tn) /Jtn/ ( T , y (r))dr(2 8 )2Discrete variable methods are commonly called numerical methods, or numerical schemes whendiscussing ODEs We adopt this convention throughout the thesis3The function / ( ) is often referred to as the increment function8
by a quadrature rule T h e classical RI form ula« used well know n quadrature rulessuch as S im p son ’s rule and were all explicitTo approxim ate (2 8 ) we choose quadrature pointsC15 c 2 5 ') cq¿1) 2,,bq.and w eightsW e th en use th e quadrature form ula9y(tn 1 ) y(tn) ]6,fc, Error(2 9)t lw ith th e derivatives approxim ated byfc, — h f9 ] a,tj kjt lE"I- c, h, ynFor RK m eth od s therefore, we have that9L iIn general this is a set o f im plicit equation s w hich w e solve for th e kt 's and u se adiscrete version o f (2 9) for our next value of y(tn i) thus9Vn l V n J 2 b' k'» 1an *12,C2a 21aT h ese im plicit R unge K u tta (IR K ) m eth od s were first introdu ced by B utcher [8 ] Ithas b ecom e standard to follow B utcher and display th e coefficients as an array thusc3«31a 32? 3 qcqO-q1a q2,,O-qqwh ,"to, OlqtoCl)(2 10)bqIt is com m on to adopt th e follow ing shorthand n otationA(2 11)w here A represents th e m atrix o f coefficients atJ, b is is th e vector o f w eights an d c isth e vector of quadrature p oints T h e quadrature p oin ts are usually called abscissae,w hile in th e literature th ey are som etim es called th e integration stages W e p oin t outth at th is representation includes th e classical exp licit form ulae if atJ 0 , w heneveri j T h en each k, is given exp licitly in term s o f th e previous ones9
It turns out however th at the nonlinear equations w hich arise from the apph cationof the I R K m ethod to (2 5 ) are ve ry expensive to solveO ne w a y to circum ventthis difficulty is to use a lower trian gu lar a rra y o f coefficientsm ethods have been called[1]atJ in ( 2 . 1 0 ), suchsem i-explicit R K m ethods m th e literature, see A lexan d era„ are equal, we have a D iago n a lly Im plicit R K (D IR K )If, in addition, all th eform ulaT h e se form ula h ave been exten sively studied b y N o rsett [52], A le xan d e r [1]and C ro u ziex [23] and h ave th e general formClaC2Û21aC3 31Û32C9a ql0,2bx¿2a(2 12)- a * KO nce again, we adopt th e sh o
outline a selection of techniques that have appeared in the literature for dealing with these problems Once again in Chapter 5 we return to numerical methods. We explain why some DAEs are solvable by numerical methods suitable for explicit ODEs and others are not We show that the index or degree of complexity of a DAE, determines both the
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