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 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