Fourth Order Four-Stage Diagonally Implicit Runge-Kutta Method for Linear Ordinary Differential Equations ABSTRACT INTRODUCTION

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1 Malasa Joural of Mathematcal Sceces (): (00) Fourth Order Four-Stage Dagoall Implct Ruge-Kutta Method for Lear Ordar Dfferetal Equatos Nur Izzat Che Jawas, Fudzah Ismal, Mohamed Sulema, 3 Azm Jaafar Departmet of Mathematcs, Facult of Scece ad Techolog, Uverst Malasa Tereggau Departmet of Mathematcs, Facult of Scece, Uverst Putra Malasa 3 Facult of Computer Scece ad Iformato Techolog Uverst Putra Malasa E-mal: urzzatjawas@ahoo.com, fudzah_@ahoo.com.m, msulema@ahoo.com.m, azm@fstm.upm.edu.m ABSTRACT A ew fourth order four-stage Dagoall Implct Ruge-Kutta (DIRK) method whch s specall desged for the tegrato of Lear Ordar Dfferetal Equatos (LODEs) s costructed. I the dervato, Butcher s error equatos are used but oe of the error equatos ca be elmated due to the propert of the LODE tself. The stablt aspect of the method s vestgated ad t s foud to have a bgger rego of stablt compared to explct Ruge-Kutta (ERK) method of the same tpe (desged for the tegrato of LODE). A set of test problems are used to valdate the method ad umercal results show that the method produced smaller global error compared to ERK method. INTRODUCTION I ths paper, we cosder the umercal tegrato of lear homogeeous sstems of ordar dfferetal equatos (ODEs) of the form = A g( x) (.) where A s a square matrx whose etres does ot deped o or x, ad ad g( x ) are vectors. Such sstems arse the umercal soluto of partal dfferetal equatos (PDEs) goverg wave ad heat pheomea after applcato of a spatal dscretzato such as fte-dfferece method. Explct Ruge-Kutta method s ver popular for smulatos of wave equatos; (see Zgg ad Chsholm (999) ad Ferraca ad Spjer (007), due to ther hgh accurac ad low memor requremets.

2 Nur Izzat Che Jawas et al. I the dervato of Ruge-Kutta (RK) methods, certa order equatos or sometmes called error equatos eed to be satsfed; see Dormad (996). These order equatos resulted from the dervatves of the fucto = f ( x, ) tself. If the fucto s lear the some of the error equatos resulted b the oleart the dervatve fucto ca be removed, thus less order equatos eed to be satsfed, hece a more effcet method some respect tha the classcal method ca be produced or derved. I ths paper, we costruct dagoall mplct Ruge-Kutta method specfcall for lear ODEs wth costat coeffcets, the the stablt aspect of the method s looed to ad a set of test equatos are used to valdate the ew method. DERIVATION OF THE METHOD We cosder the followg scalar ODE = f ( x, ) (.) Whe a geeral s-stage dagoall mplct Ruge-Kutta method s appled to the ODE, the followg equatos are obtaed, s = hb (.a) = where = f ( x c h, h a ). (.b) j= j j = s a j j= We shall alwas assume that the row-sum codto holds c, where =,.. s. Accordg to Dormad(996) ad Butcher (003), the followg eght order equatos are equatos eeded to be satsfed b fourth order four-stage DIRK method. 96 Malasa Joural of Mathematcal Sceces

3 Fourth Order Four-Stage Dagoall Implct Ruge-Kutta Method For Lear Ordar Dfferetal Equatos TABLE : Ruge-Kutta order equatos for fourth order () b τ = τ = b c () τ = bc (3) 3 τ = b a c (3) j j 6 j τ = b c ( ) 3 τ = bc a c () j j 8 j τ = b a c () 3 j j j τ = b a a c () j j j The restrcto to lear ODEs reduces the umber of equatos whch the coeffcets of the RK method must satsf Table. Zgg ad Chsholm (999) have derved ew explct RK methods whch are sutable for lear ODEs that are more effcet tha the covetoal RK methods. For ths ew fourth order DIRK method sutable for lear ODEs, equato 6 Table ca be elmated, see paper Zgg ad Chsholm (999). Ths codto s elmated b explotg the fact that, for lear ODEs, f f = = 0. x After equato 6 Table has bee elmated, we have seve equatos to be solved wth uows. So we have four free parameters whch are chose to beγ = 0.0, c = 0.05, c3 = 0.0 ad c = All the equatos Table (except equato 6) are solved usg MAPLE pacage. The coeffcets obtaed are wrtte Butcher s arra as follows: Malasa Joural of Mathematcal Sceces 97

4 Nur Izzat Che Jawas et al. c a T b Applg all the parameters to the geeral form of RK method, we have the ew fourth order four-stage DIRK method whch s sutable for lear ODEs, = where 3 = f ( x = f ( x = f ( x = f ( x h( ) 0.0h, 0.05h, 0.0h, 0.80h, h(0.0 )) h( 0.5 h( h( )) ( ) )) )) STABILITY Oe of the practcal crtera for a good method to be useful s that t must have a rego of absolute stablt. Whe a s-stage Ruge-Kutta method (equatos (.a) ad (.b)) s appled to = f ( x, ) = λ 98 Malasa Joural of Mathematcal Sceces

5 Fourth Order Four-Stage Dagoall Implct Ruge-Kutta Method For Lear Ordar Dfferetal Equatos the followg equatos are obtaed where ) = R ( hλ ( λ) = ( ˆ) = ˆ ( ˆ ) T R h R h hb I ha e ad A s (m x m), e s (m x ) are obtaed from the coeffcets of the method tself. R(hˆ ) s called the stablt polomal of the method ad for ths method t s gve as R( h) = hˆ 0.8 ˆ 0. ˆ h h ( ˆ ˆ ˆ 3 h h h ) 0.03hˆ ( 0.6 ˆ 0. ˆ ˆ 3 h h h ) 0.006hˆ. 0.8 ˆ 0. ˆ 0.03 ˆ ˆ h h h h 0.5 ˆ 0.06 ˆ ˆ ( h h h ) 0.8 ˆ 0. ˆ 0.03 ˆ ˆ h h h h 0.073( ˆ ˆ ˆ 3 h h h 0.8 ˆ 0. ˆ 0.03 ˆ ˆ h h h h ( ˆ ˆ ˆ 3 h h h ) 0.8 ˆ 0. ˆ 0.03 ˆ ˆ h h h h ( ˆ ˆ ˆ 3 h h h 0.8 ˆ 0. ˆ 0.03 ˆ ˆ h h h h 0.073( ˆ ˆ ˆ 3 h h h ). 0.8 ˆ 0. ˆ 0.03 ˆ ˆ h h h h The stablt rego s obtaed b tag R( hˆ ) = = cosθ sθ ad solve for ĥ usg Mathematca pacage (see Torrece (999)). The stablt rego for ew fourth order four-stage DIRK s show Fgure. 3 Malasa Joural of Mathematcal Sceces 99

6 Nur Izzat Che Jawas et al. Imagar Part Stablt Rego Real Part Fgure : The stablt rego for ew th order -stage DIRK We also fd the stablt polomal ad the stablt rego for the fourth order four-stage explct Ruge-Kutta method (ERK) Zgg ad Chsholm (999), ad t s show Fgure below. Imagar Part Stablt Rego Real Part Fgure : The stablt rego for th order -stage ERK 00 Malasa Joural of Mathematcal Sceces

7 Fourth Order Four-Stage Dagoall Implct Ruge-Kutta Method For Lear Ordar Dfferetal Equatos PROBLEMS TESTED AND NUMERICAL RESULTS The followg are some of the problems tested. All the problems are lear ordar dfferetal equatos. PROBLEM : '( t) = ( t) = t e 0 t, (0) = Source: Rchard L.Burde ad J.Douglas Fares (00) PROBLEM : 3t '( t) = te ( t) = te e e t 3, (0) = 0 3t 3t t Source: Rchard L.Burde ad J.Douglas Fares (00) PROBLEM 3: '( t) = ta t cos t ( t) = cos t s t 0 t, (0) = Source: J. C. Butcher (003) PROBLEM : t t '( ) = t t e ( ) = ( ) 0.5 t 0 t, (0) = 0.5 Source: Rchard L.Burde ad J.Douglas Fares (00) Malasa Joural of Mathematcal Sceces 0

8 Nur Izzat Che Jawas et al. PROBLEM 5: t t t e e t '( t) = t e t ( ) = ( ) t 5, () = 0 Source: Rchard L.Burde ad J.Douglas Fares (00) The umercal results are tabulated ad compared wth the exstg method ad below are the otatos used: H ~ Step sze used MTHD ~ Method emploed MAXE ~ Maxmum error ( x ) ~ Fourth order four-stage explct RK method Zgg ad Chsholm (999) ~ New fourth order four-stage DIRK method TABLE : Comparso betwee ad for solvg Problem... MTHD H MAXE 33e e e e-008.7e e e e-0.9e e-0.685e e e e-06 0 Malasa Joural of Mathematcal Sceces

9 Fourth Order Four-Stage Dagoall Implct Ruge-Kutta Method For Lear Ordar Dfferetal Equatos TABLE 3: Comparso betwee ad for solvg problem... MTHD H MAXE.950e e-00.e e-003 5e e e e e e e e e e-00 TABLE : Comparso betwee ad for solvg problem 3... MTHD H MAXE 9.886e e e e e e e e-0.e e e e e e-05 Malasa Joural of Mathematcal Sceces 03

10 Nur Izzat Che Jawas et al. TABLE 5: Comparso betwee ad for solvg problem... MTHD H MAXE 69800e e e e e e e e e e-0.68e e-03 85e e-05 TABLE 6: Comparso betwee ad for solvg problem 5... MTHD H MAXE 53e e e e e e e e e e e e e e-00 CONCLUSION The ew fourth order four-stage DIRK method has bee preseted for the tegrato of lear sstems of ODEs. It has a bgger stablt rego compared to explct RK method (of the same order), ad hece the formula s more stable. From the umercal results Tables to 6, we ca coclude 0 Malasa Joural of Mathematcal Sceces

11 Fourth Order Four-Stage Dagoall Implct Ruge-Kutta Method For Lear Ordar Dfferetal Equatos that the ew fourth order four-stage DIRK method whch s sutable for lear ODEs performs better terms of accurac compared to fourth order four-stage ERK method. REFERENCES Burde, R.L., Fares, J.D. 00. Numercal Aalss seveth edto, Wadsworth Group. Broos/Cole, Thomso Learg, Ic. Butcher, J.C. 00 Numercal Methods for Ordar Dfferetal Equato, Joh Wle & Sos Ltd. Dormad, J.R. 99 Numercal Methods for Dfferetal Equatos, Boca Rato, New Yor, Lodo ad Toa: CRC Press, Ic. Ferraca, L., Spjer, M.N. 00 Strog stablt of Sgl-Dagoall- Implct Ruge-Kutta methods. Report o MI 007-, Mathematcal Isttute, Lede Uverst. Torrece, B.F., Torrece, E.A How to fd the stablt regos: The Studet s Itroducto to Mathematca, Cambrdge Uverst Press: pp 3- Zgg, D.W., Chsholm T.T Ruge-Kutta methods for lear ordar dfferetal equatos, Appled Numercal Mathematcs. 3: Malasa Joural of Mathematcal Sceces 05