Multistep Runge-Kutta Methods for solving DAEs

Transcription

1 Multitep Ruge-Kutta Method for olvig DAE Heru Suhartato Faculty of Coputer Sciece, Uiverita Idoeia Kapu UI, Depok 6424, Idoeia Phoe: E-ail: Kevi Burrage Advaced Coputatioal Modellig Cetre he Uiverity of Queelad, St. Lucia, QLD 4072, Autralia e-ail : kb@ath.uq.edu.au Abtract Several ethod have bee ued by oe author to olve Differetial Algebraic Equatio (DAE), the ethod rage fro ultitep ethod uch a BDF ad EBDF ethod to oe-tep ethod uch a Ruge-Kutta ethod. I thi paper, we preet Multitep Ruge-Kutta ethod (MRK) for olvig DAE which are the exteio of our previou work for olvig ODE. We dicu fixed tepize MRK ad Variable Coefficiet MRK. he ethod treat the DAE a proble of the for M y' = f(x). We ipleet the MRK i parallel iteratio techique ad preet oe uerical tet. Keyword: DAE, Multitep Ruge Kutta Method, Variable Coefficiet, Parallel Iteratio echique.itroductio Several ethod are propoed by oe author to olve DAE i the for of '( ) ( ( )), : My x f y x f R R, y( x0) y 0 = = (.) where M i a cotat atrix, yy, 0 R, ad i the ize of the proble.he ethod rage fro ultitep ethod uch a Backward Differetiatio Forulae (BDF) ad Exteded Backward Differetiatio Forulae (EBDF), for exaple ee VODE ad DASL ad oe tep ethod uch Ruge-Kutta, for exaple ee Radau (Hairer ad Waer, 99). I thi paper, the author propoe parallel iteratio techique baed o Multitep Ruge-Kutta ethod to olve DAE. he propoed techique i the exteio of our previou work for olvig Ordiary Differetial Equatio, that i equatio (.) where M=I, for exaple ee (Burrage ad Suhartato,997, Burrage ad Suhartato, 2000, ad Suhartato, 998). 2. he core ethod he uderlyig ethod i the r-tep, -tage Multitep Ruge-Kutta (MRK) ethod of Radau type characterized by = ( A I ) y + h ( B I ) F( ), ( ) + = ( α ) + ( γ ) ( y I y h I F ) (2.)

2 Here, R repreet tage vector,,, ad are approxiatio to the olutio at the off tep poit derivative of tage vector olutio (,..., ) ad y y + r (,, ) x + ch, i=,...,, ad F( ) R repreet - i ( f ( ),..., f( ) ), y R ( ),, r repreet r previou I deote the idetity atrix of order, h i the itegratio tep, A i a -by-r atrix ad B i a -by- atrix, α = ( α,..., α r ), γ = ( γ,..., γ r ) ad i Kroecker product. he ethod i tiffly accurate if α = a, γ = b, j =,... that i y = ( e + I). j j j j Now coider the tiffly accurate ethod with fixed tepize forulae, ad defie a i (Hairer ad Waer 99), t = r +,..., t =, t = 0 r r ad chooig c with 0 < < fro i c i r 2 + = 0, i =,..., c t c c j= i j j=, j i i j lead to MRK ethod of order 2+r-2. For variable coefficiet ethod, All paraeter of the variable tepize ethod, h i (, cab, ) deped o the ratio of the tepize give by ρ = r, i, i,..., r, h = where hi = xi+ xi, i = r+,...,. By defiig where c = [ c,..., c ], q = e, q = [ q,..., q ] ( p ), j j j 0 p p p r j h e = R q = q = j = ad coiderig alo the auptio i r i= [,...,], 0, j, 2,..., r. h p p B( w) : pγ c + α q = p =,..., w C pbc Aq c p p p p ( η) : + = = 0,..., η we proved i (Burrage ad Suhartato 2000, Suhartato 998) that he axiu attaiable order of a tiffly accurate Variable coefficiet MRK ethod i 2+r-2. 2

3 Method with thee order exit atifyig C ( + r ) ad with c,..., c real, ditict, ad lyig i the iterval [ q, ). r Note that for the fixed cotat tepize ethod, the paraeter q = [0,( ),( 2),...,( r+ ) ] p p p p p q for the ethod i Note alo that i the ipleetatio of variable coefficiet ethod, we eed to copute all the paraeter of the ethod at every tep ice the paraeter deped o the ratio of the tepize. While the variable tepize ipleetatio of the fixed tepize ethod oly ue cotat ethod paraeter, but the olutio defied a = ( A I ) yˆ + h ( B I ) F( ), where yˆ y yˆ yˆ = [,,..., r+ ] ad yˆ ˆ j P xj x,..., x + ad xˆ = x ( j) h, j =,..., r +. r j 2.2 he Iteratio echique o olve = ( ), where Px ( ) i a polyoial defied o i (2.), we propoed a iteratio techique a i (Burrage ad Suhartato, 997, Suhartato, 998) that i by iteratig for exaple L tie = ( A I ) y + h ([ B W] I ) F( ( ) ( j ) + h( W I) F( ), j =,..., L ( L) + = ( I ) y e ), (2.2) where deote a iitial approxiatio or predictor to the vector, ad W i the plittig atrix. Let Z = ( A I ) y, thu the coplete iteratio i defied a ( ) For j=,,l OUER iteratio ( j ) ( j ) Copute G ( ) = h([ B W] I ) F ( ) ed For For k=,,ier INNER iteratio ( k ) ( j ) ( k ) ( I W I ) Z = [ Z G( ) h( W I ) F( Z + ( A I ) y )] ( k) ( k ) Z = Z + Z ed For j ier = Z + ( A I ) y ( ) ( ) ( ) + = ( ) y e I Several type of predictor for are coidered, thee are P which i a trivial predictor where we defie 0 P =, which i a extrapolatio predictor uig r 3

4 previou olutio, P 2 previou tage vector, vector of the lat tep, ad vector of the lat tep ad predictor 2.3 he paralleli which i a extrapolatio predictor uig r previou olutio ad P 3 P 4 which i a extrapolatio predictor uig previou tage which i a extrapolatio predictor uig previou tage y. We alo proved i (Suhartato, 998) that give a of order p, it attai order p+l after l iteratio. I (Suhartato, 998), it wa alo oted that the atrix W i (2.2) hould be i a for uch that the iteratio could be doe i parallel for each i,, i =,...,. he obviou choice of thi atrix i W = D, where D i oe diagoal atrix, hece the ( ) copoet of each vector j ca be coputed i parallel provided that there are proceor available. I the cae of the fixed-tep-ize ethod, W i eaily defied a it i fixed, available ad ued o every tep. However, thi i ot the cae for a VMRK ethod, a the coputatio of W require a certai coditio to be atified which i difficult to ipleet. For exaple, coider chooig W = D, the it i required that the pectral radiu of the atrix ( I D B) vaih i order to have good covergece for tiff proble. However, i geeral thee chee are very delicate ad very difficult to adapt to our ethod a it require ybolic coputatio of D. However, for ethod with all uber of tage, ay three, we will cotruct the plittig atrice W. I thi ipleetatio we defie the to be the triagular atrice obtaied fro the applicatio of Crout ethod o B. hi require the ue of Butcher iilarity traforatio atrice but thee ca eaily be deteried. Now ice the coputatio of each i,, i =,..., i idepedet becaue of the choice of the plitig atrix W the the proce of evaluatig the derivative, factorizatio of the iteratio atrice ad olvig the liear yte of each of the tage are doe cocurretly. 3. he iteratio for DAE Coider DAE a My'( x) = f( y( x)) where M i a cotat atrix, the coplete iteratio i Sectio 2.2 ow becoe For j=,,l OUER iteratio ( j ) ( j ) Copute G ( ) = h([ B W] I ) F ( ) For k=,,ier INNER iteratio k ( I M W I ) Z = [( I M) Z G( ed For ( ) ( j ) hw I F Z + A I y ( k ) ( ) ( ( ) )] ( k) ( k ) Z = Z + Z ) 4

5 ed For = Z + ( A I ) y ( ier) ( ) + = ( ) y e I 4. Nuerical experiet he code i ipleeted i Fortra 90. he experiet were coducted o a SGI Origi2000 yte copriig 64 R0000 MHz CPU each with 4 MB cache ituated at the Uiverity of Queelad, Autralia. We ue MRK ethod with r=3,=3 ad three proceor. We olved three proble, the firt proble i a Pedulu of idex. Figure. Pedulu, idex, =5 I the figure, the curve with tar * dot repreet the ipleetatio of the fixed tep MRK ethod, while the graph with circle 0 dot repreet the ipleetatio of a variable coefficiet MRK ethod. tep idicate the uber tep take to reach ed poit, ad cpu-tie i the tie eed to coplete the itegratio. he horizotal axi repreet the uber correct digit. It i obviou that the fixed tepize ethod i le accurate tha the variable coefficiet ethod. However, the variable coefficiet ethod require uch tie tha the fixed tepize ethod. hi i due to the tie required to copute the paraeter of the ethod for the variable coefficiet ethod. 5

6 Figure 2. Pedulu, idex 2, =5 he ecod proble i a pedulu of idex 2. It i till obviou that the fixed tepize ethod i le accurate tha the variable coefficiet ethod. However, the variable coefficiet ethod require alo the ae tie tha the fixed tepize ethod. Eve though the variable tepize ethod require tie i coputig the paraeter of the ethod, but the uber tepize take by the fixed tepize ethod are uch higher. Figure 3. Fekete Proble, idex 2, =60 he third proble i a Fekete Proble of the ize =60. Figure 3 repreet the reult of thi proble, the fixed tepize ethod i till le accurate tha the variable coefficiet ethod, ad ow the variable coefficiet ethod i fater. hi idicate that the coputatio of the ethod paraeter i le doiat i large proble. 5. Cocluio We propoed Multitep Ruge Kutta Method for olvig DAE, our experiet how that the fixed tepize ethod uffer fro accuracy, while the variable coefficiet ethod aitai it accuracy. However, the variable coefficiet 6

7 ethod have overhead cot for coputig it paraeter whe the proble ize i all. he overhead tart to le doiated whe the proble ize becoe ufficietly large. 6. Referece Burrage, K., ad Suhartato, H.,997, Parallel iterated ethod baed o Multitep Ruge-Kutta of Radau type for No Stiff Proble, Adv.Coput. Math., 7, Burrage, K., ad Suhartato, H., 997, Parallel iterated ethod baed o Multitep Ruge-Kutta of Radau type for Stiff Proble, Adv.Coput. Math., 7, Burrage, K., ad Suhartato, H., 2000, Parallel iterated ethod baed o Variable tepize Multitep Ruge-Kutta, Adv.Coput. Math.3, Hairer, E., ad Waer, G., 99, Solvig Ordiary Differetial Equatio II: Stiff ad Differetial Algebraic Proble, Spriger Serie i Cop. Math. Ed Spriger Verlag, Berli, Suhartato, H., 998,Parallel Iterated echique baed o Multitep Ruge-Kutta Method of Radau ype, PhD. hei, Departet of Matheatic, he Uiveity of Queelad, St. Lucia, QLD, Autralia 7