Solving Linear Ordinary Differential Equations using Singly Diagonally Implicit Runge-Kutta fifth order five-stage method

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1 Fudzah Ismal, Nur Izzat Che Jawas, Mohamed Suleman, Azm Jaafar Solvng Lnear Ordnar Dfferental Equatons usng Sngl Dagonall Implct Runge-Kutta ffth order fve-stage method FUDZIAH ISMAIL, NUR IZZATI CHE JAWIAS, MOHAMED SULEIMAN AND AZMI JAAFAR Department of Mathematcs, Facult of Scence Department of Informaton Sstem, Facult of Computer Scence and Informaton Technolog Unverst Putra Malasa 00, Serdang, Selangor MALAYSIA Abstract: - We constructed a new ffth order fve-stage sngl dagonall mplct Runge-Kutta (DIRK) method whch s specall desgned for the ntegratons of lnear ordnar dfferental equatons (LODEs). The restrcton to lnear ordnar dfferental equatons (ODEs) reduces the number of condtons whch the coeffcents of the Runge-Kutta method must satsf. The best strateg for practcal purposes would be to choose the coeffcents of the Runge-Kutta methods such that the error norm s mnmzed. Thus, here the error norm obtaned from the error equatons of the sxth order method s mnmzed so that the free parameters chosen are obtaned from the mnmzed error norm. The stablt aspect of the method s also looked nto and found to have substantal regon of stablt, thus t s stable. Then a set of test problems are used to valdate the method. Numercal results show that the new method s more effcent n terms of accurac compared to the exstng method. Ke-Words: - Runge-Kutta, Lnear ordnar dfferental equatons, Error norm. Introducton Man algorthms have been proposed for the numercal soluton of ntal value problem f ( x, ), ( x0 ) 0, m f : () Such algorthm s the Sngl Dagonall Implct Runge-Kutta (SDIRK) method whch was ntroduced to overcome some of the lmtatons of full mplct and explct Runge-Kutta method. Prelmnar experments have shown that these methods are usuall more effcent than the standard Sngl Implct Runge-Kutta (SIRK) method and n man cases are compettve wth backward dfferentaton formula. Ths algorthms can be used b both lnear and nonlnear sstems of ordnar equatons. However n ths paper, we consder the numercal ntegraton of lnear nhomogeneous sstems of ordnar dfferental equatons (ODEs) of the form A G(x) () where A s a square matrx whose entres does not depend on or x, and and G(x) are vectors. Such sstems arse n the numercal soluton of partal dfferental equatons (PDEs) governng wave and heat phenomena after applcaton of a spatal dscretzaton such as fnte-dfference method. Ths tpe of partal dfferental equatons can be solved numercall usng methods suggested b Rasulov and Kul [7], Rasulov et. al [8] and Zabala and Ramos []. Actuall there have been several attempts to develop effcent methods for ntegratng lnear sstems of ODEs. The basc concept of ths method s that the maor cost n evaluatng the dervatve functon s n formng the matrx A and vector G(x). ISSN: Issue 8, Volume 8, August 009

2 Fudzah Ismal, Nur Izzat Che Jawas, Mohamed Suleman, Azm Jaafar Explct Runge-Kutta method s ver popular for smulatons of wave equatons; see Zngg and Chsholm [], due to ther hgh accurac and low memor requrements. To derve Runge-Kutta (RK) methods, we need to fulfll certan order equatons; see Dormand []. These order equatons resulted from the dervatves of the functon f ( x, ) tself. If the functon s lnear then some of the error equatons resulted b the nonlneart n the dervatve functon can be removed, thus less order equatons need to be satsfed, hence a more effcent method n some respect than the classcal method can be derved. In ths paper, we construct dagonall mplct Runge-Kutta method specfcall for lnear ODEs wth constant coeffcents. We consder the prncpal terms of the local truncaton error to mnmze the error norm. Then, a few test equatons are used to valdate the new method. Materals and Methods. Dervaton of the method In ths secton, we consder the followng scalar ODE f ( x, ) () When a general s-stage dagonall mplct Runge-Kutta method s appled to the ODE, the followng equatons are obtaned, Chsholm []. The order equatons are elmnated b explotng the fact that, for lnear ODEs, f x f 0. x Zngg and Chsholm [] too have derved a new explct RK methods whch are sutable for lnear ODEs that are more effcent than the conventonal RK methods. Table : Order equatons for ffth order Runge-Kutta method sutable for LODEs. () b. () b c. () b c. () a c b 6 5. () b c 6. () a c b where k n s h b k () n f ( x c h, h a k ) () n n 7. () a a c b k k k 8. (5) b c 0 We shall alwas assume that the row-sum s condton holds c a, where,..s. Accordng to Dormand [], there are 7 order equatons (error equatons) needed to be satsfed b the ffth order fve-stage RK method. The restrcton to lnear ODEs reduces the number of equatons whch the coeffcents of the RK method must satsf see Zngg and 9. (5) a c 5 6 b 0 0. (5) a a c 8 b k k 0 k. (5) a a a c 9 b k km m 0 km ISSN: Issue 8, Volume 8, August 009

3 Fudzah Ismal, Nur Izzat Che Jawas, Mohamed Suleman, Azm Jaafar Usng the smplfng assumpton: b a b c ),,...,5 (5) We have ( 5. a a a c 9 b k km m 60 km 6. a a a a c 0 b k km mn n 70 kmn b a c ( 6 b c ) ( b c ), thus equaton can be removed, smlarl we can remove equatons 6 and 9 n table. Thus, usng (5) the order equatons are replaced b smpler equatons. The are: b a ba b5a5 b ( c ba b5a5 b ( c b5a5 b ( c 5 c5 ) ) ) Altogether there are equatons needed to be satsfed and we have 5 unknowns. So, we can have four free parameters whch are chosen to be c, c, c and. Solvng whch, we have all equatons n terms of c, c c and., The order equatons for the sxth order method are the order equatons n table and the addtonal order equatons gven n table. as obtaned b Zngg and Chsholm []. Table : Addtonal order equatons for sxth order Runge-Kutta method 5. c b 6. a c 7 b 0. a a c 5 b k k 0 km In order to choose the free parameters c, c, c and, the prncpal terms of the local truncaton error must be consdered. Usng the error functon p n p ( p) ( p) and RK error coeffcents [], the prncpal term for ffth order method s 5 6 F F The best strateg for practcal purposes would be to choose the free RK parameters s to mnmze the error norm, see Dormand []; ( p) ( p) n p ( ( p) So we have the prncpal error norm for ths method; ( ) ( 7 ) ( 5 ) ( 9 ) ( 0) where are the error equatons assocated wth the sxth order method, (n table ). Substtutng the free parameters nto A, we obtaned the prncpal error norm n terms of c,c and., c Mnmzng the error norm, we have c , c , c and Substtutng the values of c, c, c and and solvng all the equatons we fnall get all the coeffcents of the new SDIRK method for LODEs as follows; ) ISSN: Issue 8, Volume 8, August 009

4 Fudzah Ismal, Nur Izzat Che Jawas, Mohamed Suleman, Azm Jaafar c c c c a a a a f ( x, ) R( h ) R( hˆ) hb ˆ ( I ha ˆ ) T where A s (m x m), e s (m x ) are obtaned from the method tself and R(hˆ) s called the stablt polnomal of the method. The stablt regon s obtaned b takng R( hˆ) cos sn. Usng the MATHEMATICA package we obtaned the stablt polnomal and also the stablt regon. The stablt polnomal for new ffth order fve-stage SDIRK method s e a a a a a a b b b b b c a a a a a55. Stablt One of the practcal crtera for a good method to be useful s that t must have regon of absolute stablt. When an s-stage Runge-Kutta method (equatons () and ()) s appled to the test equaton, R (hˆ) hˆ 0.00hˆ 0.06hˆ 0.00hˆ hˆ hˆ 0.066hˆ 0.007hˆ hˆ hˆ hˆ 0.008hˆ 0.000hˆ hˆ hˆ 0.00hˆ hˆ hˆ hˆ hˆ hˆ hˆ 0.05hˆ 0.007hˆ hˆ hˆ 0.095hˆ 0.00hˆ hˆ hˆ 0.058hˆ hˆ 0.000hˆ hˆ 0.06hˆ hˆ hˆ hˆ 0.07hˆ 0.05hˆ 0.000hˆ hˆ hˆ 0.070hˆ 0.005hˆ wth value of ; ISSN: Issue 8, Volume 8, August 009

5 Fudzah Ismal, Nur Izzat Che Jawas, Mohamed Suleman, Azm Jaafar where ˆ ˆ h h ˆ h ˆ ˆ5 h h. The stablt polnomal s solve for ĥ whch gves the value of R ( h ˆ) ; ths s done b usng Mathematca package (see Torrence []). The stablt regon s obtaned b tracng the values of ĥ and s shown n Fgure. Where the vertcal axs s the magnar part and the horzontal axs s the real part. where R ( hˆ) hˆ( hˆ 0.0 ( 0.08 hˆ ˆ ˆ ˆ h ) 0.7 (0.0 h 0.7 h ) ˆ ˆ (0.7 h h ) ( ˆ ˆ ˆ h h h ) 0.70 (0.60 ˆ 0.08 ˆ ˆ h h h ) 0.70 ( ˆ ˆ ˆ h h h ˆ h )) ImagnarPart 7.5 StabltRegon Equatng ˆ) (h R cos sn and solvng for h we have the stablt regon of the method. 5.5 Imagnar Part StabltRegon Real Part - Fgu re : The stablt regon for the 5 th order 5-stage SDIRK method Real Part The stablt analss for ffth order fve-stage explct Runge-Kutta () method whch has been derved b Zngg and Chsholm [] s dscussed below; Where the stablt polnomal s obtaned as n the SDIRK method and the stablt polnomal for the explct method s denoted b R h, Fgure 5.: The stablt regon for method Clearl the mplct method has bgger regon of stablt compared to the explct method and hence more stable. ISSN: Issue 8, Volume 8, August 009

6 Fudzah Ismal, Nur Izzat Che Jawas, Mohamed Suleman, Azm Jaafar Results and Dscusson We use the method to obtan the numercal solutons to the followng problems, all of them are lnear ODEs. Exact Soluton: PROLEM : ' ( t) tan t ( t) cos t sn t Source: J. C. utcher [ ] PROLEM : t t '( ) t e t t ( t) t ( e e) t 5, () 0 Source: urden and Fares []. PROLEM : ( 0), (0) 0, [0,0] Exact Soluton: cos t 0 t, (0) Source: Tam [0] PROL EM 5: 5 ( 0) 0, (0) 0 (0), [0, ] Exact Soluton: ( x) cos( x) 6sn( x) 6x ( x) sn( x) 6 cos( x) 6 ( x) sn( x) cos( x) Source: Flowers [6] PROLEM : Source: Suleman [9] PR OLEM 6: ( 0) 0, (0), [0,5] ISSN: Issue 8, Volume 8, August 009

7 Fudzah Ismal, Nur Izzat Che Jawas, Mohamed Suleman, Azm Jaafar Exact Soluto n: ( x) xe x T able : Numercal results for problem ( x) ( x) e x MTHD H MAXE Source: ronson [] The numercal results are tabulated and compared wth the exstng method and below are the notatons used: H ~ Step sze used MTHD ~ Method emploed MAXE ~Maxmum error The true soluton mnus the computed soluton ( x ) : ~ New ffth order fvestage SDIRK method wth mnmzed error norm for LODEs. ~ Ffth order fve-stage explct RK method for LODEs (Zngg and Chsholm, []) SDIRK(II)~ Optmal fourth order fve-stage SDIRK (Ferracna and Spker, [5]) e-009. ERK 5.75e e e-0. ERK e e e-0. ERK e e e-0. ERK e e e ERK e e e-0 6. ERK e e e e e-0 ISSN: Issue 8, Volume 8, August 009

8 Fudzah Ismal, Nur Izzat Che Jawas, Mohamed Suleman, Azm Jaafar Ta ble : Numercal results for pro blem MTHD MAXE.695e e e e e e e e-00.05e e e-00.80e e e-00.98e e e e e e-00.09e e-009.9e-006.6e-009 Ta ble 6: Numercal resu lts for pro blem e e e-009 MTHD H. 0. MAXE.0e e e e-00.0e e e e e e-00.97e e e e-00.58e-00 Table 5: Numercal results for problem MTHD H MAXE.060e e e e e e-0.05e e-0.5e e e e e e e e e e e e e-0 ISSN: Issue 8, Volume 8, August 009

9 Fudzah Ismal, Nur Izzat Che Jawas, Mohamed Suleman, Azm Jaafar Table 7: Numercal results for proble m 5 MTHD H MAXE e-0.066e e L SDIRK(II).0e e e e e-0.876e e e-008 SDIRK(II) 5.866e SDIRKL e-0.77e e e-0 L e-00 SDIRK(II).58e e e e SDIRK5l.067e L.896e-0 SDIRK(II) 9.70e-0.05e L.009e-0 SDIRK(II).5e e e-05 SDIRK(II) e e e-06 SDIRK(II) 6.888e-0 Concluson The new ffth order fve-stage SDIRK method wth mnmzed error norm has been presented for the ntegraton of lnear ODEs. It has a substantal regon of stablt, thus, t s stable. From the numercal results gven n Table -8, and for all the problems tested, we can conclude that the new ffth order fve-stage SDIRK method whch s sutable for lnear ODEs performs better n terms of maxmum error compared to the ffth order fve-stage ERK method and the optmal fourth order fve-stage SDIRK method. Table 8: Numercal results for problem 6 H MAXE MTHD.6e e e e e e e e e-007 References [] ronson R. Modern Introductor Dfferental Equaton, Schaum s Outlne Seres. USA: McGraw-Hll 97. [] urden R.L., Fares J.D. Numercal Analss seventh edton, Wadsworth Group. rooks/cole, Thomson Learnng, Inc. 00 [] utcher J.C. Numercal Methods for Ordnar Dfferental Equaton, John Wle & Sons Ltd. 00 [] Dormand J.R. Numercal Methods for Dfferental Equatons, oca Raton, New York, London and Toko: CRC Press, Inc ISSN: Issue 8, Volume 8, August 009

10 Fudzah Ismal, Nur Izzat Che Jawas, Mohamed Suleman, Azm Jaafar [6] Flowers,. H An Introducton to Numercal Methods n C+ +, New York: Unverst Oxford Press 000. [7] Rasulov, R and Kul, R. H. Numercal Soluton of One Dmensonal Nonlnear Wave Equaton wth Twce Nonlneart n a class of Dscontnuous Functons, WSEAS Transactons on Mathematcs, vol 5, no, 006, : 6-8. [8] Rasulov, M, Snsoal,. and Hata, S. Numercal Smulaton and Intal-oundar Value Problems for Traffc Flow n a class of Dscontnuous Functons, WSEAS Transactons on Mathematcs, vol 5, no, 006, : 9-. [9] Suleman M.. Solvng Hgher Order ODEs Drectl b Drect Integraton Method, Appled Math. And Computaton. (), 989: [0] Tam, H. W. Two-stage Parallel Methods for the Numercal Soluton of Ordnar Dfferental Equatons, Sam J. Sc. Stat. Comput. (5, 99: [] Torrence.F., Torrence E.A.: How to fnd the stablt regons, The Student s Introducton to Mathematca, 999, pp. -6. [] Zabala, D. and Ramos, A. L. Effect of the Fnte Dfference Soluton Scheme n a Free oundar Convectve Mass Transfer Model, WSEAS Transactons on Mathematcs, vol 6, no 6, 007, : [5] Ferracna L., Spker M.N. Strong stablt of Sngl-Dagonall-Implct Runge-Kutta methods. Report no MI 007-, 007, Mathematcal Insttute, Leden Unverst. [] Zngg D.W., Chsholm T.T. Runge- Kutta methods for lnear ordnar dfferental equatons, Appled Numercal Mathematcs., 999, pp ISSN: Issue 8, Volume 8, August 009