Một số phương pháp song song dạng runge kutta giải bài toán không cương

I HC QUC GIA H NI TRNG I HC KHOA HC T NHIấN NGUYN THU THY MT S PHNG PHP SONG SONG DNG RUNGE - KUTTA GII BI TON KHễNG CNG LUN N TIN S TON HC H NI - 2014 I HC QUC GIA H NI TRNG I HC KHOA HC T NHIấN NGUYN THU THY MT S PHNG PHP SONG SONG DNG RUNGE - KUTTA GII BI TON KHễNG CNG Chuyờn ngnh: Toỏn hc tớnh toỏn Mó s: 62 46 30 01 LUN N TIN S TON HC Ngi hng dn khoa hc: GS.TSKH Nguyn Hu Cụng H NI - 2014 LI CAM OAN Tụi xin cam oan õy l cụng trỡnh nghiờn cu ca riờng tụi Cỏc kt qu nờu lun ỏn l trung thc v cha tng c cụng b bt k cụng trỡnh no khỏc Tỏc gi Nguyn Thu Thy LI CM N Lun ỏn c hon thnh di s hng dn ca GS TSKH Nguyn Hu Cụng Thy ó dn dt tỏc gi lm quen vi nghiờn cu khoa hc t tỏc gi ang l hc viờn cao hc Ngoi nhng ch dn v mt khoa hc, s ng viờn v lũng tin tng ca thy dnh cho tỏc gi luụn l ng lc ln giỳp tỏc gi t tin v say mờ nghiờn cu Qua õy tỏc gi xin by t s bit n sõu sc v lũng quý mn i vi thy Tỏc gi cng xin c by t lũng bit n n cỏc thy cụ v cỏc bn ng nghip xemina B mụn Toỏn hc tớnh toỏn, trng i hc Khoa hc T nhiờn-i hc Quc Gia H Ni ó to mụi trng hc v nghiờn cu thun li giỳp tỏc gi honh thnh lun ỏn ny Ti õy tỏc gi ó nhn c nhiu ch dn, gúp ý cng nh mt mụi trng nghiờn cu sụi ni v thõn thin, iu khụng th thiu quỏ trỡnh nghiờn cu, hon thnh lun ỏn ca tỏc gi Tỏc gi xin gi li cỏm n ti cỏc thy cụ khoa Toỏn-C-Tin hc, Phũng Sau i hc, Trng i hc Khoa hc T nhiờn- i hc Quc Gia H Ni, ni tỏc gi ó hc v nghiờn cu Tỏc gi xin c by t lũng bit n n Ban Giỏm hiu, Ban ch nhim khoa Toỏn-Tin v B mụn Toỏn ng dng trng i hc S phm H Ni ó to nhng iu kin thun li quỏ trỡnh tỏc gi hc tp, cụng tỏc v hon thnh lun ỏn ny Trong quỏ trỡnh hc v hon thnh lun ỏn, tỏc gi ó nhn c s quan tõm giỳp v gúp ý ca GS.TSKH Phm K Anh, PGS.TSKH V Hong Linh, Tỏc gi xin chõn thnh cm n cỏc Giỏo s v s giỳp quý bỏu ny Cui cựng, tỏc gi xin c by t lũng bit n n ụng b, b m, anh ch em hai bờn ni ngoi, cựng chng v bn bố ó gúp ý v ng viờn tỏc gi quỏ trỡnh hc v hon thnh lun ỏn Tỏc gi MC LC MC LC DANH MC CC T VIT TT MT S K HIU CHUNG M U Chng MT S KIN THC C S 1.1 11 Phng phỏp Runge-Kutta 12 1.1.1 Cp chớnh xỏc ca phng phỏp Runge-Kutta 14 1.1.2 Tớnh n nh ca phng phỏp Runge-Kutta 15 1.2 Cỏc phng phỏp Runge-Kutta hin 16 1.3 Cỏc phng phỏp Runge-Kutta n 18 1.4 Phng phỏp Runge-Kutta lp song song (PIRK) 21 1.4.1 Ni dung phng phỏp PIRK 23 1.4.2 Cp chớnh xỏc ca phng phỏp PIRK 24 1.5 1.4.3 S n nh ca phng phỏp PIRK 24 1.4.4 S hi t ca quỏ trỡnh lp 26 Mt s mó tớnh toỏn tun t 26 1.5.1 Phng phỏp kp thờm cú cp chớnh xỏc - mó DOPRI5 1.5.2 Phng phỏp kp thờm cú cp chớnh xỏc 8- mó DOPRI853 28 Phng phỏp ngoi suy- mó ODEX 31 Ba bi toỏn th 37 1.5.3 1.6 27 Chng PHNG PHP LP SONG SONG DNG RUNGEKUTTA HAI BC MT DA TRấN CC IM TRNG KHP GAUSS-LEGENDRE 2.1 2.2 40 Phng phỏp dng Runge-Kutta hai bc mt da trờn cỏc im trựng khp Gauss-Legendre 41 2.1.1 n nh tuyn tớnh 44 2.1.2 Th nghim s 49 Phng phỏp lp song song dng Runge-Kutta hai bc mt da trờn cỏc im trựng khp Gauss-Legendre 50 2.2.1 iu kin bc 52 2.2.2 S hi t ca quỏ trỡnh lp 54 2.2.3 Min n nh 55 2.2.4 Th nghim s 57 2.2.5 So sỏnh vi cỏc phng phỏp song song 59 2.2.6 So sỏnh vi cỏc mó tun t 62 Chng PHNG PHP LP SONG SONG GI RUNGE-KUTTA HAI BC VI CHIN LC IU KHIN BC LI 3.1 3.2 3.3 65 Phng phỏp gi Runge-Kutta hai bc kp thờm vi bc li thay i 66 3.1.1 iu kin bc 68 3.1.2 Cụng thc kp thờm 72 Phng phỏp PIPTRK vi chin lc iu khin bc li 73 3.2.1 iu kin bc cho cụng thc d bỏo 75 3.2.2 S hi t ca quỏ trỡnh lp 77 3.2.3 iu khin bc li 77 Th nghim s 79 3.3.1 Xỏc lp phng phỏp PIPTRKSC 79 3.3.2 So sỏnh vi cỏc mó song song 81 3.3.3 So sỏnh vi cỏc mó tun t 83 3.3.4 Tớnh hiu qu ca chin lc iu khin bc li 85 Chng PHNG PHP GI RUNGE-KUTTA BA BC 4.1 4.2 89 Phng phỏp gi Runge-Kutta ba bc (EPThRK) 90 4.1.1 iu kin bc 92 4.1.2 Tớnh n nh 97 Cỏc th nghim s 98 4.2.1 Chn phng phỏp EPThRK 98 4.2.2 So sỏnh vi cỏc mó song song 100 4.2.3 So sỏnh vi cỏc mó tun t 102 4.2.4 So sỏnh phng phỏp EPThRK vi phng phỏp TBTPIRKG v PIPTRKSC 104 KT LUN KIN NGH MT S HNG NGHIấN CU TIP THEO 108 109 DANH MC CễNG TRèNH KHOA HC CA TC GI LIấN QUAN N LUN N 110 TI LIU THAM KHO 111 MT S K HIU CHUNG Mt s kớ hiu thụng thng Rd khụng gian cỏc vộc t thc d chiu C s phc C s phc vi phn thc khụng dng Vi s phc z C, Re(z), Im(z) ln lt l phn thc v phn o ca s phc z (A) l ph ca ma trn A (A) l bỏn kớnh ph ca ma trn A Ly tha ca mt vộc t Gi s c = (c1 , c2 , , cs )T , ú ck = (ck1 , ck2 , , cks )T Toỏn t exp( d ) dx d d d2 dn exp( ) = + + + ããã + + dx dx 2!dx n!dxn Kớ hiu vộc t e Vộc t e luụn hiu l vộc t cú tt c cỏc thnh phn bng Vộc t hm Gi s f (x, y) l hm thc ca hai bin x, y Nu thay x v y tng ng bi hai vộc t v = (v1 , v2 , , vs )T v w = (w1 , w2 , , ws )T thỡ ta c vộc t hm vi s thnh phn: f (v, w) = [f (v1 , w1 ), f (v2 , w2 ), , f (vs , ws )]T Nu x R, cũn y thay bi w = (w1 , w2 , , ws )T thỡ ta cú: f (x, w) = [f (x, w1 ), f (x, w2 ), , f (x, ws )]T DANH MC CC T VIT TT EPThRK Explicit pseudo three-step Runge-Kutta method Phng phỏp gi Runge-Kutta ba bc ERK Explicit Runge-Kutta Runge-Kutta hin IRK Implicit Runge-Kutta Rungge-Kutta n PC Predictor-Corrector D bỏo-Hiu chnh PIPTRK parallel-iterated pseudo two-step Runge- Kutta methods Phng phỏp lp song song gi Runge-Kutta hai bc PIPTRKSC Parallel-iterated pseudo two-step Runge-Kutta method with step size control Phng phỏp lp song song gi Runge-Kutta hai bc vi chin lc iu khin bc li PTRK Pseudo two-step RK methods Phng phỏp gi Runge-Kutta hai bc TBTIRKG Two-step-by-two-step IRK methods based on Gauss-Legendre collocations points Phng phỏp dng Runge-Kutta n hai bc mt da trờn cỏc im trựng khp Gauss-Legendre TBTRKG Two-step-by-two-step Runge-Kutta-type corrector methods based on Gauss-Legendre collocation points Phng phỏp hiu chnh dng Runge-Kutta hai bc mt da trờn im trựng khp Gauss-Legendre TBTPIRKG two-step-by-two-step parallel-iterated Runge-Kutta-type PC methods based on Gauss-Legendre collocation points Phng phỏp lp song song dng Runge-Kutta hai bc mt da trờn cỏc im trựng khp Gauss-Legendre 111 TI LIU THAM KHO [1] Phm K Anh (2008), Gii tớch s, Nh xut bn i hc Quc Gia H Ni [2] Nguyn Hu Cụng (2002), Cỏc phng phỏp song song dng RungeKutta- Nystrăom, Nh xut bn i hc Quc Gia H Ni [3] Lờ Ngc Xuõn (2007), Mt s phng phỏp song song gii h phng trỡnh vi phõn, Lun ỏn Tin s Toỏn hc, i hc Khoa hc T nhiờn- i hc Quc gia H Ni Ting Anh [4] Bellen, A., Vermiglio, R., Zennaro, M (1990), "Parallel ODE-solvers with stepsize control", J Comput Appl Math 31, pp.277-293 [5] K Burrage (1993), "Efficient block predictor-corrector methods with a small number of corrections", J Comput Appl Math 45, pp.139-150 [6] K Burrage (1993), "Parallel methods for initial value problems", Appl Numer Math 11, pp.5-25 [7] K Burrage (1995), Parallel and Sequential Methods for Ordinary Differential Equations, Clarendon Press, Oxford [8] K Burrage and H Suhartanto (1997), "Parallel iterated methods based on multistep Runge-Kutta mehods of Radau type", Advances in Computational Mathematics 7, pp.37-57 [9] K Burrage (1978), "A special family of Runge-Kutta methods for solving stiff differential equations", BIT 18, pp.22-41 112 [10] J.C Butcher (1963), "Coefficients for the study of Runge-Kutta Integration Processes", J of the Australian Math Soc., 3, pp.185201 [11] J.C Butcher (1964), "Implicit Runge-Kutta processes", Math Comp 18, pp.50-64 [12] J.C Butcher (1964), "Integration processes based on Radau quadrature formulas", Math Comp 18, pp.233-244 [13] J.C Butcher (1964), "On Runge-Kutta processes of high order", J of the Australian Math Soc 4, pp.179-194 [14] J.C Butcher (1964), "On the attainable order of Runge-Kutta methods", Math Comp 19, pp.408-417 [15] J.C Butcher (1985), "The non-existence of ten stage eighth order explicit Runge-Kutta methods", BIT 27, pp.521-540 [16] J.C Butcher (1977), "A-stable implicit Runge-Kutta methods", BIT 17, pp.375-378 [17] J.C Butcher (1987), The Numerial Analysys of Ordinary Differential Equations, Runge-Kutta and General Linear Methods, Wiley, New York [18] N.H Cong (1994), "Parallel iteration of symmetric Runge-Kutta methods for nonstiff initial-value problems", J Comput Appl Math 51, pp.117-125 [19] N.H Cong (1999), "Explicit pseudo two-step Runge-Kutta methods for parallel computers", Int J Comput Math 73, pp.77-91 [20] N.H Cong (1999), "Continuous variable stepsize explicit pseudo two-step RK methods", J Comput Appl Math 101, pp.105-116 113 [21] N.H Cong and T Mitsui (1996), "Collocation-based two-step Runge-Kutta methods", Japan J Indust Appl Math 13, pp.171183 [22] N.H Cong and T Mitsui (1997), "A class of explicit parallel twostep Runge-Kutta methods", Japan J Indust Appl Math 14, pp.303-313 [23] N.H Cong and T Mitsui (2003), "Parallel PC iteration of pseudo two-step RK methods for nonstiff IVPs", Japan J Indust Appl Math 20, pp.51-64 [24] N.H Cong, H Podhaisky and R Weiner (1998), "Numerical experiments with some explicit pseudo two-step RK methods on a shared memory computer", Comput Math Appl 36, pp.107-116 [25] N.H Cong and H.T Vi (1995), "An improvement for explicit parallel Runge-Kutta methods", Vietnam J Math 23, pp.241-252 [26] N.H Cong and L.N Xuan (2003), "Parallel-iterated RK-type PC methods with continuous output formulas", Int J Comput Math 80, pp.1027-1037 [27] N.H Cong and L.N Xuan (2003), "Parallel-iterated RK-type PC methods with continuous output formulas", Int J Comput Math 23, pp.241-252 [28] N.H Cong and L.N Xuan (2008), "Improved parallel-iterated pseudo two-step RK methods for nonstiff IVPs", Appl Numer Math 58, pp.160-170 [29] N.H Cong and L.N Xuan (2008), "twostep-by-twostep PIRK-type PC methods with continous output formulas", J Comput Appl Math 221, pp.165-173 114 [30] N.H Cong and N.T Thuy (2011), "Two-step-by-two-step PIRKtype PC methods based on Gauss-Legendre collocation points", J Comput Appl Math 236, pp.225-233 [31] N.H Cong and N.T Thuy (2012), Stability of Two-Step-by-TwoStep IRK methods based on Gauss-Legendre collocation points and an application, Vietnam Journal of Mathematics, 40, no.1, pp.115126 [32] N.H Cong and N.T Thuy (2014), "A class of explicit pseudo threestep Runge-Kutta methods", (submitted) [33] N.H Cong and N.T Thuy (2014), "Parallel iterated pseudo twostep RK methods with stepsize control", Japan Journal of Industrial and Applied Mathematics, 31, no 2, pp 441-460 [34] N.H Cong (2001), "A general family of pseudo two-step RungeKutta methods", SEA Bull Math 25, pp.61-73 [35] N.H Cong, "Explicit pseudo three-step Runge-Kutta-Nystrăom methods", in preparation [36] A.R Curtis (1975)," High-order explicit Runge-Kutta formulae , their uses and limitations", J Inst Math Appl 16, pp.35-55 [37] A.R Curtis (1964), Tables of Jacobian Elliptic Functions Whose Arguments are Rational Fractions of the Quater Period, H.M.S.O., London [38] Gear, C.W (1988), "Parallel methods for ordinary differential equations", Calcolo 25, pp 1-20 [39] E Hairer (1978), "A Runge-Kutta method of order 10", J Inst Math Appl 21, pp.47-59 115 [40] E Hairer, S.P Nứrsett and G Wanner (1987), Solving Ordinary Differential Equations I Nonstiff Problems, 1st edition, SpringerVerlag, Berlin [41] E Hairer, S.P Nứrsett and G Wanner (1993), Solving Ordinary Differential Equations I Nonstiff Problems, 2nd edition, SpringerVerlag, Berlin [42] E Hairer and G Wanner (1991), Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin [43] P.J van der Houwen and N.H Cong (1993), "Parallel block predictor-corrector methods of Runge-Kutta type", Appl Numer Math 13, pp.109-123 [44] P.J van der Houwen and B.P Sommeijer (1990), "Parallel iteration of high-order Runge-Kutta methods with stepsize control", J Comput Appl Math 29, pp.111-127 [45] P.J van der Houwen and B.P Sommeijer (1992), "Block RungeKutta methods on parallel computers", Z Angew Math Mech 68, pp.3-10 [46] P.J van der Houwen, and B.P Sommeijer (1991), "Iterated RungeKutta methods on parallel computers", SIAM J Sci Stat Comput 12, pp.1000-1028 [47] P.J van der Houwen, B.P Sommeijer and W Couzy (1992), "Embedded diagonally implicit Runge-Kutta algorithms on parallel computers", Math Comput 58, pp.135-159 [48] T.E Hull, W.H Enright, B.M Fellen and A.E Sedgwick (1972), "Comparing numerical methods for ordinary differential equations", SIAM J Numer Anal 9, pp.603-637 116 [49] G.Yu Kulikov and R Weiner (2010), "Variable-stepsize interpolating explicit parallel peer methods with inherent global error control", SIAM J Sci Comput 32, pp.1695-1723 [50] G.Yu Kulikov and R Weiner (2010), "Doubly quasi-consistent parallel explicit peer methods with built-in global error estimation", J Comput Appl Math 233, pp.2351-2364 [51] S.P Nứrsett and H.H Simonsen (1989), "Aspects of parallel RungeKutta methods, in Numerical Methods for Ordinary Differential Equations", Proceedings LAquilla 1987, Lecture Notes in Mathematics, 1386, (Edited by A Bellen, C.W Gear and E Russo), Springer-Verlag, Berlin [52] B.A Schmitt, R Weiner and S Jebens (2009), "Parameter optimization for explicit parallel peer two-step methods", Appl Numer Math 59, pp.769-782 [53] B.A Schmitt and R Weiner (2010), "Parallel start for explicit parallel two-step peer methods", Numer Algorithms 53, pp.363-381 [54] L.F Shampine and M.K Gordon (1975), Computer Solution of Ordinary Differential Equations, The Initial Value Problems, W.H Freeman and Company, San Francisco [55] O Axelsson (1969), "A class of A-stable methods", BIT 9, pp.185199 [56] K Dekker and J.G Verwer (1984), Stability of Runge-Kutta Mehtods for Stiff Nonlinear Differential Equations, North-Holland, Amsterdam [57] P Kaps (1981), Rosenbrock-type methods, in: Numerical Methods for Stiff Initial Value Problems, G Dahlquist and R Jeltsch (eds.), 117 Bericht Nr 9, Inst fă ur Geometrie und Praktische Mathematik der RWTH Aachen [58] J.D Lambert (1991), Numerical Methods for Ordinary Differential Systems, The Initial Value Problems, John Wiley & Sons [59] B.P Sommeijer, W Couzy and P.J van der Houwen (1992), "Astable parallel block methods for ordinary and integro-differential equations", Appl Numer Math 9, pp.267-281 [60] O Widlund (1967), "A note on unconditionally stable linear multistep methods", BIT 7, pp.65-70 [61] M.T Chu and H Hamilton (1987), "Parallel solution of ODEs by multi-block methods", SIAM J Sci Statist Comput 3, pp.342-353 [62] R Weiner, G.Yu Kulikov and H Podhaisky (2012), "Variablestepsize doubly quasi-consistent parallel explicit peer methods with global error control", Appl Numer Math 62, pp.1591-1603