On error estimation in almost Runge-Kutta (ARK) methods

Transcription

1 Joural of Applied Mathematics & Bioiformatics, vol.3, o., 3, 5-35 ISSN: prit), olie) Sciepress Ltd, 3 O error estimatio i almost Ruge-Kutta ARK) methods Ochoche Abraham, Kaode R. Adeboe ad Alimi K. Olumide 3 Abstract I Ruge-Kutta Theor, it is commo kowledge that the bouds for the local trucatio errors do ot form a suitable basis for moitorig the local trucatio error 7]. I a umber of literatures icludig ], Richardso Etrapolatio techique has bee show to be a ver reliable meas of obtaiig error estimates for Ruge-Kutta methods. I this paper, we have sought to ivestigate, b meas of rigorous umerical eperimets, if this effectiveess of Richardso etrapolatio techique eteds to ARK methods ad our results show that it does t. Mathematics Subject Classificatio: 65G99 Departmet of Iformatio ad Media Techolog, Federal Uiversit of Techolog Mia, Nigeria, abochoche@futmia.edu.g Departmet of Mathematics ad Statistics, Federal Uiversit of Techolog Mia, Nigeria, profadeboe@gmail.com 3 Departmet of Mathematics ad Statistics, Federal Uiversit of Techolog Mia. Nigeria, alimikaum@ahoo.com Article Ifo: Received : Februar, 3. Revised : March 6, 3 Published olie : Jue 3, 3

2 6 O error estimatio i almost ARK methods Kewords: Error estimatio, Almost Ruge-Kutta, Richardso etrapolatio, Eact, Approimate Itroductio Mathematics is the ligua fraca of life; b it the phsical world fids epressio. Historicall, differet equatios have origiated i Chemistr, Phsics ad Egieerig. More recetl the have also arise i models i Medicie, Biolog, Athropolog ad the likes. I this paper, we shall restrict our attetio to just umerical methods for approimatig the solutio of ordiar differetial equatios with prime focus o iitial value problems IVP); so called because the coditios o the solutio of the differetial equatio are all specified at the start of the trajector i.e. the are iitial coditios. Numerical solutio of ODE is the most importat techique ever developed i cotiuous time damics. Sice most ODEs are ot soluble aalticall, umerical itegratio is the ol wa to obtai iformatio about the trajector. Ma differet methods have bee proposed ad used i a attempt to solve accuratel, various tpes of ODE. However, there is a hadful of methods kow ad used uiversall e.g. Ruge-Kutta 6], ]), Adam-Bashforth-Moulto 3], 8]) ad Backward Differece Formulae). All these, discretize the differetial sstem, to produce a differece equatio or map. The methods produce differet maps from the same equatio, but the have the same aim: that the damics of the maps, should correspod closel, to the damics of the differetial equatio. From the Ruge-Kutta famil of algorithms, come the most well-kow ad used methods for umerical itegratio 5]. With the advet of computers, umerical methods are ow a icreasigl attractive ad efficiet wa to obtai approimate solutios to differetial equatios that have hitherto proved difficult, eve impossible to solve aalticall.

3 Ochoche Abraham, K.R. Adeboe ad A.K. Olumide 7 Prelimiar Notes Almost Ruge-Kutta ARK) methods are a ver special class of geeral liear methods itroduced b Butcher i 997 4]. The basic idea of these methods is to retai the multi-stage ature of Ruge-Kutta methods, while allowig more tha oe value to be passed from step to step. Hece, the have a multi-value ature. These methods have advatages over traditioal methods, which are to be foud i low-cost local error estimatio ad dese output. These latter features will be a cosequece of the higher stage orders that are possible because of the multi-value ature of the ew methods. Though this multivalue ature brigs its ow difficulties i the sese that satisfactor solutios have to be foud ot ol for stepsize chages, but also for the startig steps. The umber of values passed betwee steps varies amog geeral liear methods; the umber of values is three for ARK methods. Of the three iput ad output values i ARK methods, oe approimates the solutio value i.e. ) the secod approimates the scaled first derivative h ) ) while the third approimates the scaled secod derivatives h ) ). To simplif the startig procedure, the secod derivative is required to be accurate ol to withi O h 3 ), where h as usual is the stepsize. I other to esure that this low order does ot adversel affect the solutio value, the method has ibuilt aihilatio coditios. As a result of these etra iput values, we are able to obtai stage order two, ulike eplicit Ruge-Kutta methods that are ol able to obtai stage order oe. The mai advatage of the icrease i stage order is that error estimates ad cotiuous solutios ca be more easil achieved ]. ARK have the geeral form:

4 8 O error estimatio i almost ARK methods.) ) ) ) ] ] ] ] 3 ] ] = r s s hf hf hf V B U A with ). = β β T T s T e b b Ac c Ae c e A V B U A where s is the umber of iteral stages. The use of a ARK method is ver similar to that of a RK method, with the mai differece beig that three pieces of iformatio is ow passed betwee steps. The first two startig values are ) ad )) hf respectivel ad the third startig value is obtaied b takig a sigle Euler step forward ad the takig the differece betwee the derivatives at these two poits. Therefore, the startig vector is give b ) ] ).3 )) )) ) )), ), hf hf hf hf + This choice of startig method was chose for its simplicit, but it is adequate, at least for low order methods. The method for computig the three startig approimatios ca be writte i the form of the geeralized Ruge Kutta tableau

5 Ochoche Abraham, K.R. Adeboe ad A.K. Olumide 9 where the zero i the first colum of the last two rows idicates the fact that the term is abset from the output approimatio. This ca be iterpreted i the same wa as a Ruge Kutta method, but with three output approimatios. The usual wa to chage the stepsize is to simpl scale the vector i the same wa oe would scale a Nordsieck vector 9]. Settig r = h j h j meas the vector eeds to be scaled b r. r For a order p method the three output values are give b the equatios below ] p+ = ) + Oh ), ] p+ = h ) + O h ),.4) ] 3 3 = h ) + Oh ). I choosig the coefficiets of the method, we are careful to esure that the simple stabilit properties of Ruge-Kutta methods are retaied ]. I this paper we will cocetrate o methods where A is strictl lower triagular, ad hece the methods are eplicit, but most of the theor ca be carried over to implicit methods ]. 3 Richardso etrapolatio techique The deferred approach to the limit, otherwise kow as Richardso etrapolatio ] ivolves solvig a problem twice usig step sizes h ad h. Uder the localizig assumptio that o previous errors have bee made, we ma write = T = h + h 3.) p+ p+ + ) + + ϕ, )) O ) where p is the order of the method, ϕ, )) h p + is the pricipal local trucatio error. Net, we will compute +, a secod approimatio to + ), *

6 3 O error estimatio i almost ARK methods obtaied b applig the same method at with stepleght h. Uder the same localizig assumptio, it follows that: ) = ϕ, )) h) + O h ) 3.) * p+ p+ + + ad o epadig ϕ, )) about, ), ) = ϕ, )) h) + O h ) 3.3) * p+ p+ + + O subtractig 3.) from 3.3), we obtai ) = ) ϕ, )) h + O h ) * p+ p+ p+ + + Therefore, the pricipal local trucatio error which is take as a estimate for the local trucatio error ma be writte as: ) * p+ + + = + = p+ ϕ, )) h T ) = * + + T + p+ 3.4) 3.5) 3. Numerical eperimets I ] as well as other literature, Richardso Etrapolatio has bee show to be a meas of obtaiig quick ad acceptable estimates of the local trucatio errors i computatios usig a s-stage eplicit Ruge-Kutta method, without havig to obtai the eact solutio first. The goal of this paper is to ivestigate the viabilit of equatio 3.5) for ARK methods b solvig the followig iitial value problems: t e. = ); ) =. Eact solutio: = t/ e. = cos, ) =. Eact Solutio: si = Ce, C =. A umerical solver was developed usig Java programmig laguage to solve the differetial equatios above usig the followig Almost Ruge-Kutta methods.

7 Ochoche Abraham, K.R. Adeboe ad A.K. Olumide Method ARK45: A Fourth order method with five stages T c =,,,,], ϕ = Method ARK5: A Fifth order method with five stages 5 / / / 5 / 989 / / 3 6 / 9 97 / / / / / / / 95 4 / 55 4 / 55 6 / /5 7/ 65 43/ / / / / 97 5 / 8 57 / 593 / / 6

8 3 O error estimatio i almost ARK methods 4 Mai Results I Figure we ca see from the solutio curves that the curve for the error estimates usig Richardso etrapolatio is ver far from the curve of the umerical approimatio usig ARK45 with h=. ad h =.. I Figure we ca see from the solutio curves agai, that the curve for the error estimates usig Richardso etrapolatio is ver far from the curve of the umerical approimatio usig ARK5 with h=. ad h =.. I Figure 3 we ca see from the solutio curves that the estimates from Richardso etrapolatio agai, has o similarities whatsoever with the solutio curves of the umerical approimatio usig ARK45 with h =. ad h =.. I Figure 4 just as i Figure 3, we ca see from the solutio curves, that the estimates from Richardso etrapolatio agai, has o similarities whatsoever with the solutio curves of the umerical approimatio usig ARK5 with h =. ad h =.. 5 Labels of figures ad tables errors Figure : Graph of actual errors ad estimated errors for problem usig ARK45

9 Ochoche Abraham, K.R. Adeboe ad A.K. Olumide 33 Figure : Graph of actual errors ad estimated errors for problem usig ARK5 Figure 3: Graph of actual errors ad estimated errors for problem usig ARK45

10 34 O error estimatio i almost ARK methods Figure 4: Graph of actual errors ad estimated errors for problem usig ARK5 6 Coclusio From the results obtaied i sectio 5., it ca be therefore cocluded that while Richardso etrapolatio techique is a ver effective meas of obtaiig acceptable error estimates for Ruge-Kutta methods, it is ot the same for Almost Ruge-Kutta methods. Ackowledgemets. The Authors would like to ackowledge the immese cotributios ad support of Prof. Joh C. Butcher of the Uiversit of Aucklad, New Zealad, who ispired ad ecouraged Dr. Abraham to carr out this ivestigative stud as far back as 9 while the were sittig, sippig coffee at the Aucklad Iteratioal Airport, waitig for Dr. Abraham s flight to Sigapore to start boardig. The result is this paper; a collaborative research work.

11 Ochoche Abraham, K.R. Adeboe ad A.K. Olumide 35 Refereces ] O. Abraham ad G. Bolari, O Error Estimatio i Ruge-Kutta Methods, Leoardo Joural of Scieces, 8, Jauar-Jue, ), -. ] O. Abraham, Developmet of Some New Classes of Eplicit Almost Ruge-Kutta EARK) Methods for No-Stiff Differetial Equatios, Ph.D Thesis, Federal Uiversit of Techolog, Mia, Nigeria,. 3] F. Bashforth ad J.C. Adams, A attempt to Test the Theories of Capillar Actio b Comparig the Theoretical ad Measured Forms of Drops of Fluid, with a Eplaatio of the Method of Itegratio Emploed i Costructig the Tables which Give the Theoretical Forms of Such Drops, Cambridge Uiversit Press, Cambridge, ] J.C. Butcher, A Itroductio to Almost Ruge Kutta Methods, Applied Numerical Mathematics, 4, 997), ] J.H.C. Jula ad O. Piro, The Damics of Ruge-Kutta Methods, Iteratioal Joural Bifurcatio ad Chaos,, 99), -4. 6] W. Kutta, Beitrag fur N Aherugsweise Itegratio Totaler Differetialgleichuge, Z. Math. Phs., 46, 9), ] J.D. Lambert, Numerical Methods for Ordiar Differetial Sstems, Joh Wile ad Sos, USA, pp. 49-5, 99. 8] F.R. Moulto, New Methods i Eterior Ballistics, Uiversit of Chicago Press, 96. 9] A. Nordseick, O Numerical Itegratio of Ordiar Differetial Equatios, Math. Comp., 6, 96), -49. ] N. Rattebur, Almost Ruge-Kutta Methods for Stiff ad No-Stiff Problems, PhD Thesis, The Uiversit of Aucklad, New Zealad, 5. ] L.F. Richardso, The deferred approach to the limit, -sigle lattice, Tras. Ro. Soc. Lodo, 6, 97), ] C. Ruge, Ueber Die Nuumerische Auflösug vo Differetialgleichuge, Math. A., 46, 895),