В.Б. Андреев - Численные методы (2 в 1). (2007) (1160465), страница 20
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Îí òàêæå ÿâëÿåòñÿ îäíîýòàïíûì ìåòîäîì ÐóíãåÊóòòû.Èññëåäóåì òåïåðü íàèáîëåå öåëåñîîáðàçíûé âûáîð ïàðàìåòðîâ b1 è a11 ñ òî÷êèçðåíèÿ ìèíèìèçàöèè ïîãðåøíîñòè àïïðîêñèìàöèè. ×òîáû íàéòè ïîãðåøíîñòü àïïðîêñèìàöèè, ïåðåïèøåì óðàâíåíèå (16.11) â âèäåun+1 − un= b1 f (Y1 )τ(16.12)(ñð. ñ (15.7), (15.8), (15.10) è (15.12)), à ðåøåíèå óðàâíåíèÿ (16.10) îáîçíà÷èì ÷åðåçY1 (un ). Åñëè, êàê è âûøå, zn = un − u(tn ), òîzn+1 − znu(tn+1 ) − u(tn )= b1 f (Y1 (u(tn ) + zn )) −.ττÈ ñíîâà, ðàñêëàäûâàÿ ïåðâîå ñëàãàåìîå ïðàâîé ÷àñòè ïî ôîðìóëå Òåéëîðà, íàõîäèì,÷òî·¸zn+1 − zn∂fu(tn+1 ) − u(tn )= b1 f (Y1 (u(tn ))) +(ũ)zn −=τ∂uτ∂f ∂Y1= b1(ũ)zn + ψn ,∂Y1 ∂uãäåu(tn+1 ) − u(tn )(16.13)ψn = b1 f (Y1 (u(tn ))) −τ ïîãðåøíîñòü àïïðîêñèìàöèè, à Y1 (u(tn )) ðåøåíèå óðàâíåíèÿ (16.10) ñ u(tn ) âìåñòîun , ò.å.Y1 (u(tn )) = u(tn ) + τ a11 f (Y1 (u(tn ))).(16.14)Çàìå÷àíèå 16.2.
Ïîãðåøíîñòü àïïðîêñèìàöèè (16.13) ïðåäñòàâëÿåò ñîáîé ðàçíîñòüìåæäó ïðàâîé è ëåâîé ÷àñòÿìè óðàâíåíèÿ (16.12), åñëè òóäà âìåñòî ïðèáëèæåííîãîðåøåíèÿ ïîäñòàâèòü òî÷íîå (ñð. ñ çàìå÷àíèåì 15.3).16.2. ÎÄÍÎÝÒÀÏÍÛÅ ÌÅÒÎÄÛ ÐÓÍÃÅ-ÊÓÒÒÛ159Ðàçëîæèì ïîãðåøíîñòü àïïðîêñèìàöèè (16.13) ïî ñòåïåíÿì τ . Èìååì"#2fd f (Y1 ) ¯¯τ 2 dgψn = b1+τ+−τ =0τ =0dτ2 dτ 2·¸τ 00τ 2 ˜0000− u (tn ) + u (tn ) + ũ .26¯f (Y1 ) ¯Èç (16.14) íàõîäèì, ÷òî Y1 |τ =0 = u(tn ) è, ñëåäîâàòåëüíî,¯f (Y1 ) ¯τ =0 = f (u(tn )).Ñíîâà ñ èñïîëüçîâàíèåì (16.14)df (Y1 ) ¯¯df dY1 ¯¯df==(u(tn ))a11 f (u(tn )),τ=0τ=0dτdY1 dτduà èç óðàâíåíèÿ (16.10 )u0 (tn ) = f (u(tn )),u00 (tn ) =dfdfdudf(u(tn )) =(u(tn )) (tn ) =f.dtdudtduÏîýòîìó·¸1dfψn = (b1 − 1)f (u(tn )) + τ b1 a11 −f (u(tn )) (u(tn )) + O(τ 2 ).2duÒåì ñàìûì, äëÿ òîãî, ÷òîáû ïîãðåøíîñòü àïïðîêñèìàöèè áûëà O(τ 2 ), íåîáõîäèìî èäîñòàòî÷íî, ÷òîáû âûïîëíÿëèñü óñëîâèÿb1 = 1,a11 b1 = 1/2.(16.15)Îòñþäà íàõîäèìb1 = 1,a11 = 1/2è, ñëåäîâàòåëüíî, íåÿâíûé îäíîýòàïíûé ìåòîä Ðóíãå-ÊóòòûτY1 = un + f (Y1 ),2un+1 = un + τ f (Y1 )(16.16)èìååò âòîðîé ïîðÿäîê àïïðîêñèìàöèè.Çàìå÷àíèå 16.3.
Èç ïåðâîãî óðàâíåíèÿ (16.16) ñëåäóåò, ÷òî ìîìåíò âðåìåíè, íàêîòîðûé Y1 ïðèáëèæàåò u(t), åñòü t + τ /2, èáî äëÿ çàäà÷è u0 = 1, u(0) = 0, èìåþùåéðåøåíèå u = t, Y1 = un + τ /2 = tn + τ /2.160 16. ÌÅÒÎÄÛ ÐÓÍÃÅ-ÊÓÒÒÛÑîîòíîøåíèÿ (16.16) ìîæíî ïðåîáðàçîâàòü. Èñêëþ÷èâ f (Y1 ), íàéäåì, ÷òî un+1 =2Y1 − un . Âûðàæàÿ îòñþäà Y1 è ïîäñòàâëÿÿ åãî âî âòîðîå óðàâíåíèå (16.16), ïîëó÷è쵶un+1 + unun+1 = un + τ f.2Çàìå÷àíèå 16.4. Ìåòîä (16.16) î÷åíü ñèëüíî íàïîìèíàåò ìåòîä Ðóíãå (15.10), (15.11).Îòëè÷èå ìåæäó íèìè ñîñòîèò â òîì, ÷òî çäåñü ïðîìåæóòî÷íîå çíà÷åíèå íàõîäèòñÿ ïîíåÿâíîé ôîðìóëå, à â ìåòîäå Ðóíãå ïî ÿâíîé ôîðìóëå (15.11). Ìåòîä (16.16), êàêìû óæå ñêàçàëè, ÿâëÿåòñÿ îäíîýòàïíûì (íåÿâíûì) ìåòîäîì Ðóíãå-Êóòòû, à ìåòîä(15.10), (15.11) äâóõýòàïíûì (ÿâíûì) ìåòîäîì.
Ïîä÷åðêíåì, ÷òî ñëîâó ýòàï çäåñüìû ïðèäàåì ÷åòêèé ìàòåìàòè÷åñêèé ñìûñë.16.3 Ìåòîäû òðåòüåãî ïîðÿäêà àïïðîêñèìàöèèÂûÿñíèì îãðàíè÷åíèÿ íà êîýôôèöèåíòû (16.9), îáåñïå÷èâàþùèå òðåòèé ïîðÿäîê àïïðîêñèìàöèè s-ýòàïíîãî ìåòîäà Ðóíãå-Êóòòû. Äëÿ ýòîãî íóæíî èññëåäîâàòü ïîãðåøíîñòü àïïðîêñèìàöèèψn := ψn (τ ) :=sXbi f (Yi (u(tn ))) −i=1ãäåYi (u(tn )) = u(tn ) + τsXu(tn+1 ) − u(tn ),τaij f (Yj (u(tn ))) =: Yi (u(tn ); τ ).(16.17)j=1Ðàñêëàäûâàÿ ψn (τ ) ïî τ äî òðåòüåãî ïîðÿäêà, áóäåì èìåòü¯¯¯·¸s2 2X¯¯¯df(Y)τdf(Y)ii3¯¯ψn (τ ) =bi f (Yi ) ¯¯++O(τ ) −+τ¯¯2dτ2dττ=0τ=0τ=0¸·i=1τ 2 000τ 0003− u (tn ) + u (tn ) + u (tn ) + O(τ ) .26(16.18)Ïîñêîëüêó f (Yi (u(tn ))) åñòü ñëîæíàÿ ôóíêöèÿ τ , òî âû÷èñëèì ñíà÷àëà ïðîèçâîäíûåïî τ ôóíêöèè Yi (u(tn ); τ ) ïðè τ = 0.
Èç (16.17) ñ ó÷åòîì (16.9), íàõîäèì, ÷òYi ¯¯ = u(tn ),τ =0" s#¯¯¯sXX¯¯¯dYdfi ¯Yi0 ¯¯ ==aij f (Yi ) + τaijYj0 ¯¯ = f (u(tn ))ci ,¯d τ τ =0d Yjτ =0τ =0j=1j=1"#¯¯ssss2XXX¯¯dfdf Xdfdf0 200 ¯aijYi00 ¯¯ = 2=2f(u(t))Yj0 +τaij(Y)+τaYaij cj .nijjj ¯2dYdYdYdujjjτ =0τ=0j=1j=1j=1j=116.3. ÌÅÒÎÄÛ ÒÐÅÒÜÅÃÎ ÏÎÐßÄÊÀ ÀÏÏÐÎÊÑÈÌÀÖÈÈ161Òåïåðü ìîæíî íàéòè ïðîèçâîäíûå f :¯¯f (Yi (u(tn ))) ¯¯ = f (u(tn )),τ =0¯¯df (Yi ) ¯¯df 0 ¯¯df=Y=f(u(t))ci ,nidτ ¯τ =0 dYi ¯τ =0du¯· 2¸¯µ ¶2 Xsd f 0 2 df 00 ¯¯d2 f 2d2 f (Yi ) ¯¯df2==f(u(t))Yc+2f(u(t))aij cj .(Y)+nndτ 2 ¯dY 2 idYi i ¯du2 iduiτ =0τ =0j=1Äàëåå, èç (16.10 )u0 = f,u00 =df 0u = f 0 f,duu000 = f 00 u0 f + (f 0 )2 u0 = f 00 f 2 + (f 0 )2 fè, ñëåäîâàòåëüíî,¢u(tn+1 ) − u(tn )ττ 2 ¡ 00 2= f + f 0f +f f + (f 0 )2 f + O(τ 3 ).τ26Ïîäñòàâëÿÿ òåïåðü íàéäåííûå ðàçëîæåíèÿ â (16.18), áóäåì èìåòüψn =·sXi=1"biτ2f + τ f f 0 ci +2Ãf 2 f 00 c2i + 2f f 02sX!#aij cjj=1¸¢ττ ¡ 2 00− f + ff0 +f f + f f 02 + O(τ 3 ).262−(16.19)Îòñþäà, ïðèðàâíèâàÿ êîýôôèöèåíòû ïðè îäèíàêîâûõ ñòåïåíÿõ τ , íàõîäèì, ÷òî óñëîâèÿ òðåòüåãî ïîðÿäêà àïïðîêñèìàöèè ñóòüsXi=1sXi=1sXi=1bi = 1 ,1bi ci =2(16.20),1bi c2i = ,3sX1bi aij cj = .6i,j=1Ïðè ýòîì (16.20) ñóòü óñëîâèÿ âòîðîãî ïîðÿäêà àïïðîêñèìàöèè.(16.21)162 16.
ÌÅÒÎÄÛ ÐÓÍÃÅ-ÊÓÒÒÛÇàìå÷àíèå 16.5. ×òîáû èìåòü óñëîâèÿ ÷åòâåðòîãî ïîðÿäêà àïïðîêñèìàöèè, ê óñëîâèÿì (16.20), (16.21) íóæíî äîáàâèòü ñëåäóþùèå óñëîâèÿ:sXi=11bi c3i = ,4sX1bi ci aij cj = ,8i,j=1sXbi aij c2ji,j=1sX(16.22)1= ,12bi aij ajk ck =i,j,k=11.24Çàìå÷àíèå 16.6.
Óñëîâèÿ (16.20) ñ ó÷åòîì çàìå÷àíèÿ 16.1 ìîæíî òðàêòîâàòü êàêóñëîâèÿ òî÷íîñòè êâàäðàòóðíîé ôîðìóëû èç (16.6) íà ëèíåéíûõ ôóíêöèÿõ.1 Äîáàâëåíèå ê ýòèì óñëîâèÿì ïåðâîãî èç ñîîòíîøåíèé (16.21), à çàòåì è ïåðâîãî èç ñîîòíîøåíèé(16.22) íà óêàçàííóþ êâàäðàòóðíóþ ôîðìóëó íàêëàäûâàåò äîïîëíèòåëüíûå óñëîâèÿòî÷íîñòè íà êâàäðàòè÷íûõ è êóáè÷íûõ ôóíêöèÿõ.Óïðàæíåíèå 16.1. Ïîêàçàòü, ÷òî ìåòîä òðàïåöèé (15.12) ÿâëÿåòñÿ íåÿâíûì äâóõ-ýòàïíûì ìåòîäîì Ðóíãå-Êóòòû âòîðîãî ïîðÿäêà àïïðîêñèìàöèè. (Íàéòè âñå bi , aij èïîêàçàòü íåâûïîëíåíèå õîòÿ áû îäíî èç óñëîâèé (16.21))Îòâåò:00011/2 1/21/2 1/2µ1 2 1 20 + 122¶16= .3Óïðàæíåíèå 16.2. Ïîêàçàòü, ÷òî ìåòîä Ðóíãå (15.10), (15.11) ÿâëÿåòñÿ ÿâíûìäâóõýòàïíûì ìåòîäîì Ðóíãå-Êóòòû âòîðîãî ïîðÿäêà.Îòâåò:1/21/201·¸110·0+1·6=.4316.4 Äâóõýòàïíûå íåÿâíûå ìåòîäû òðåòüåãî ïîðÿäêàÏîëîæèì â (16.20), (16.21) ïàðàìåòð s = 2.
 ðåçóëüòàòå ñèñòåìà ïðèìåò âèäb1 + b2 = 1,c1 b1 + c2 b2 = 1/2,c21 b1 + c22 b2 = 1/3,b1 (a11 c1 + a12 c2 ) + b2 (a21 c1 + a22 c2 ) = 1/6.1 Âåäüci = θi , ò.å. êîîðäèíàòà ïåðåìåííîé èíòåãðèðîâàíèÿ â i-îì óçëå.(16.23)16.4. ÄÂÓÕÝÒÀÏÍÛÅ ÍÅßÂÍÛÅ ÌÅÒÎÄÛ ÒÐÅÒÜÅÃÎ ÏÎÐßÄÊÀ163Ýòà ñèñòåìà ñîäåðæèò ÷åòûðå óðàâíåíèÿ è øåñòü íåèçâåñòíûõ (Åñëè íå ñ÷èòàòü c1è c2 , çàäàâàåìûå (16.9)). Ïîýòîìó, âîîáùå ãîâîðÿ, äâà èç ýòèõ íåèçâåñòíûõ äîëæíûîñòàòüñÿ ñâîáîäíûìè, à îñòàëüíûå âûðàçèòüñÿ ÷åðåç íèõ.
Ñèñòåìà (16.23) íåëèíåéíàÿ,è íåò ðåãóëÿðíûõ ñïîñîáîâ åå ðåøåíèÿ. Óêàæåì îäèí ïóòü, ïðèâîäÿùèé ê ðåøåíèþýòîé ñèñòåìû.Äëÿ îòûñêàíèÿ ðåøåíèÿ ñèñòåìû (16.23) ïðåäïîëîæèì ñíà÷àëà, ÷òî íåèçâåñòíûåc1 è c2 íàéäåíû, è ðàññìîòðèì ïåðâûå òðè óðàâíåíèÿ (16.23) êàê ñèñòåìó ëèíåéíûõóðàâíåíèé îòíîñèòåëüíî b1 è b2 . Ïîñêîëüêó ýòà ñèñòåìà ïåðåîïðåäåëåíà, òî äëÿ ååðàçðåøèìîñòè íåîáõîäèìî îáðàùåíèå â íóëü îïðåäåëèòåëÿ ðàñøèðåííîé ìàòðèöû¯¯¯1 1 1 ¯¯¯¯c1 c2 1/2¯ = 1 c2 + 1 c21 + c1 c22 − c21 c2 − 1 c22 − 1 c1 =¯¯ 2 2223¯c1 c2 1/3¯ 311= (c2 − c1 ) − (c2 + c1 )(c2 − c1 ) + c1 c2 (c2 − c1 ) =3· 2¸1 c1 + c2= (c2 − c1 )−+ c1 c2 = 0.32(16.24)Ïðîàíàëèçèðóåì ýòî ñîîòíîøåíèå.
Åñëè áû c1 = c2 , òî ïîñëåäíåå óðàâíåíèå (16.23)ïðèíÿëî áû âèäc21 b1 + c22 b2 = 1/6,÷òî ïðîòèâîðå÷èò òðåòüåìó óðàâíåíèþ (16.23), è ïîýòîìóc1 − c2 6= 0.(16.25)Òåì ñàìûì, èç (16.24) ñëåäóåò, ÷òî2 − 3(c1 + c2 ) + 6c1 c2 = 0èëè(3 − 6c1 )c2 = 2 − 3c1 .Ïîñêîëüêó c1 = 1/2 íå óäîâëåòâîðÿåò ýòîìó óðàâíåíèþ, òîc1 6= 1/2è ìîæíî íàéòèc2 =2 − 3c1.3 − 6c1(16.26)(16.27)Ðàçðåøèì òåïåðü ïåðâûå äâà óðàâíåíèÿ (16.23) îòíîñèòåëüíî b1 è b2 ïðè ïîìîùèïðàâèëà Êðàìåðà. Áóäåì èìåòü¯¯¯¯¯¯¯1 1 ¯¯ 1 1¯¯1 1¯¯ = 1/2 − c1¯ = c2 − 1/2,¯ = c2 − c1 6= 0,∆2 = ¯¯∆1 = ¯¯∆ = ¯¯c1 1/2¯1/2 c2 ¯c1 c2 ¯164 16. ÌÅÒÎÄÛ ÐÓÍÃÅ-ÊÓÒÒÛè, ñëåäîâàòåëüíî,b1 =c2 − 1/21,=2c2 − c14(3c1 − 3c1 + 1)b2 =1/2 − c1.c2 − c1(16.28)Èç (16.27), (16.28) ñëåäóåò, ÷òî c1 ìîæíî ïðèíÿòü çà ïàðàìåòð.
 êà÷åñòâå âòîðîãîïàðàìåòðà âîçüìåì a12 . Òîãäàa11 = c1 − a12 .(16.29)Ïîñêîëüêó(16.30)a21 = c2 − a22 ,òî, ïîäñòàâëÿÿ ýòè âûðàæåíèÿ äëÿ a11 è a21 â ïîñëåäíåå èç óðàâíåíèé (16.23), ïîëó÷èìb1 [(c1 − a12 )c1 + a12 c2 ] + b2 [(c2 − a22 )c1 + a22 c2 ] = 1/6.Ïðèíèìàÿ âî âíèìàíèå (16.28), âòîðîå èç óðàâíåíèé (16.23) è ðàçðåøàÿ ïîëó÷åííîåñîîòíîøåíèå îòíîñèòåëüíî a22 , áóäåì èìåòüa22 =1/6 − c1 /2 − a12 (c2 − 1/2)(1 − 3c1 )(1 − 2c1 ) − a12=.1/2 − c13(1 − 2c1 )2(16.31)Ñîîòíîøåíèÿ (16.27), (16.28), (16.29), (16.30), (16.31) çàäàþò äâóõïàðàìåòðè÷åñêîåñåìåéñòâî íåÿâíûõ äâóõýòàïíûõ ìåòîäîâ Ðóíãå-Êóòòû òðåòüåãî ïîðÿäêà.Åñëè ïîëîæèòü, íàïðèìåð,a12 = 0,c1 ≡ a11 = a22 ,(16.32)2òî äëÿ√ c1 èç (16.30) ïîëó÷èì êâàäðàòíîå óðàâíåíèå 6c1 − 6c1 + 1 ñ êîðíÿìè c1 = γ =3± 3.
Òàáëèöà Áóò÷åðà äëÿ ýòîãî ìåòîäà èìååò âèä6θ1 = γθ2 = 1 − γγ01 − 2γ γ1/21/2√3± 3γ=.6(16.33)Óïðàæíåíèå 16.3. Äîêàçàòü, ÷òî ìåòîä (16.33) åñòü ìåòîä (16.27)-(16.32).16.5 ßâíûå äâóõýòàïíûå ìåòîäû ñèëó îïðåäåëåíèÿ äëÿ ÿâíîãî äâóõýòàïíîãî ìåòîäà a11 = a12 = a22 = 0 è ëèøüa21 6= 0. (Ïðè a21 = 0 ìû ïîëó÷èì ÿâíûé îäíîýòàïíûé ìåòîä.) Ïîñêîëüêó äâóõýòàïíûåìåòîäû òðåòüåãî ïîðÿäêà èìåþò ëèøü äâà ñâîáîäíûõ ïàðàìåòðà, à ìû çàäàëè òðè, òîðàññ÷èòûâàòü íà òðåòèé ïîðÿäîê ó ÿâíûõ äâóõýòàïíûõ ìåòîäîâ, âîîáùå ãîâîðÿ, íåïðèõîäèòñÿ.
Ìû ïîêàæåì, ÷òî òàê îíî è åñòü.16.6. ÄÂÓÕÝÒÀÏÍÛÉ ÌÅÒÎÄ ×ÅÒÂÅÐÒÎÃÎ ÏÎÐßÄÊÀ165Ïðèíèìàÿ a21 çà ïàðàìåòð, èç óñëîâèé âòîðîãî ïîðÿäêà àïïðîêñèìàöèè (16.20),êîòîðûå â íàøåì ñëó÷àå ïðèíèìàþò âèäb1 + b2 = 1,íàõîäèìµb1 =11−2a21a21 b2 = 1/2,¶,b2 =1.2a21Òåì ñàìûì, ÿâíûå äâóõýòàïíûå ìåòîäû Ðóíãå-Êóòòû âòîðîãî ïðîÿäêà îáðàçóþò îäíîïàðàìåòðè÷åñêîå ñåìåéñòâî.Äàëåå, ïîñêîëüêó â ðàññìàòðèâàåìîì ñëó÷àå íàðÿäó ñ a11 , a12 , a22 è c1 = 0, òî ëåâàÿ÷àñòü ÷åòâåðòîãî èç óñëîâèé (16.23) îáðàùàåòñÿ â íóëü è ñëåäîâàòåëüíî ýòî óñëîâèåâûïîëíåííûì áûòü íå ìîæåò.
Ìû äîêàçàëè, ÷òî ÿâíûõ äâóõýòàïíûõ ìåòîäîâ òðåòüåãîïîðÿäêà íå ñóùåñòâóåò.Åñëè ïîëîæèòü, íàïðèìåð, a21 = 1, òî ïîëó÷èì ìåòîä ÕîéíàY 1 = un ,un+1Y2 = un + τ f (Y1 ),τ= un + [f (Y1 ) + f (Y2 )].20001101/2 1/2Óïðàæíåíèå 16.4. Âûïèñàòü âñå ïîñòðîåííûå ìåòîäû âòîðîãî ïîðÿäêà.16.6 Äâóõýòàïíûé ìåòîä ÷åòâåðòîãî ïîðÿäêàÊîýôôèöèåíòû ìåòîäà ÷åòâåðòîãî ïîðÿäêà äîëæíû óäîâëåòâîðÿòü åùå ÷åòûðåì óñëîâèÿì (16.22). Õîòÿ ó äâóõýòàïíîãî ìåòîäà òðåòüåãî ïîðÿäêà îñòàëîñü òîëüêî äâà ïàðàìåòðà, ñóùåñòâóåò åäèíñòâåííûé äâóõýòàïíûé ìåòîä ÷åòâåðòîãî ïîðÿäêà. Åãî êîýôôèöèåíòû ñóòü√√1/2 − 3/61/41/4 − 3/6√√(16.34)1/2 + 3/6 1/4 + 3/61/41/21/2R1Çàìå÷àíèå 16.7.
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