В.Б. Андреев - Численные методы (1113834), страница 22
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òàê æå êàê è äëÿ äèôôåðåíöèàëüíîãî óðàâíåíèÿ. Îáóñëîâëåíî ýòî òåì, ÷òî â êàæäîì óðàâíåíèè ñâÿçàíû ìåæäóñîáîé çíà÷åíèÿ ïðèáëèæåííîãî ðåøåíèÿ â äâóõ ñîñåäíèõ óçëàõ ñåòêè (íå ñ÷èòàÿ ïðîìåæóòî÷íûõ çíà÷åíèé). Äðóãîé êëàññ ìåòîäîâ ñîñòàâëÿþò òàê íàçûâàåìûå ìíîãîøàãîâûå ìåòîäû, â êîòîðûõ óðàâíåíèÿ ñâÿçûâàþò çíà÷åíèÿ ïðèáëèæåííîãî ðåøåíèÿ âíåñêîëüêèõ ñîñåäíèõ óçëàõ.17.1 Ìåòîäû ÀäàìñàÍàèáîëåå èçâåñòíûìè èç ìíîãîøàãîâûõ ìåòîäîâ è íàèáîëåå ñòàðûìè ÿâëÿþòñÿ ìåòîäûÀäàìñà. Îïèøåì ýòè ìåòîäû íà ïðèìåðå óðàâíåíèÿ (17.1). Âíîâü áóäåì ïðåäïîëàãàòü,÷òî íà îòðåçêå èíòåãðèðîâàíèÿ ââåäåíà ðàâíîìåðíàÿ ñåòêà ñ øàãîì τ , à óðàâíåíèå(17.1) ïðîèíòåãðèðîâàíî ïî îòðåçêó ìåæäó óçëàìè tn è tn+1Z tn+1u(tn+1 ) − u(tn ) =f (u(t))dt.(17.2)tnÇàìåíèì ïîäûíòåãðàëüíóþ ôóíêöèþ â (17.2) èíòåðïîëÿöèîííûì ìíîãî÷ëåíîì Ëàãðàíæà ïî íåêîòîðûì óçëàì ñåòêè ω (à íå ïî ïðîìåæóòî÷íûì óçëàì, êàê ýòî áûëî âìåòîäàõ Ðóíãå-Êóòòû (!)).
 çàâèñèìîñòè îò òîãî, ó÷àñòâóåò ëè óçåë tn+1 â èíòåðïîëÿöèè f (u(t)) èëè íåò, ðàçëè÷àþò íåÿâíûå è ÿâíûå ìåòîäû Àäàìñà.à) ßâíûå ìåòîäû Àäàìñà. Ïðåäïîëîæèì, ÷òî u(t) èçâåñòíà â k óçëàõ ñåòêè ωtn , tn−1 , . . . , tn+1−k .173(17.3)174 17. ËÈÍÅÉÍÛÅ ÌÍÎÃÎØÀÃÎÂÛÅ ÌÅÒÎÄÛÏîñòðîèì ïî ýòèì óçëàì èíòåðïîëÿöèîííûé ìíîãî÷ëåí Ëàãðàíæà ñòåïåíè k − 1 äëÿïîäûíòåãðàëüíîé ôóíêöèè f (u(t)) èç (17.2)f (u(t)) ≈ Lk−1 (t) =kXpj (t)f (u(tn+1−j )),(17.4)j=1ãäå, êàê îáû÷íî,pj (t) =kYi=1i6=jt − tn+1−itn+1−j − tn+1−i(17.5)ñóòü âåñîâûå ôóíêöèè èíòåðïîëÿöèîííîãî ïîëèíîìà (ìíîãî÷ëåíû ñòåïåíè (k − 1)),îáðàùàþùèåñÿ â íóëü ïðè t = tn+1−i , i = 1, j − 1, j + 1, k è â åäèíèöó ïðè t = tn+1−j ).Ïîäñòàâëÿÿ (17.4), (17.5) â (17.2) è çàìåíÿÿ ïðèáëèæåííîå ðàâåíñòâî íà òî÷íîå, ïîëó÷èì ñëåäóþùåå óðàâíåíèå äëÿ îïðåäåëåíèÿ ïðèáëèæåííîãî ðåøåíèÿun+1 − un = τkXbj f (un+1−j ),(17.6)j=1ãäå1bj =τZZtn+11p̂j (θ)dθ =pj (t)dt =tnZ100kYθ−1+idθ.i−ji=1(17.7)i6=jÎïðåäåëåíèå 17.1.
×èñëåííûé ìåòîä (17.6), (17.7) íàçûâàåòñÿ ÿâíûì k -øàãîâûììåòîäîì Àäàìñà (èíîãäà åãî íàçûâàþò ìåòîäîì Àäàìñà-Áýøôîðòà).Ïðèìåðû. 1◦ . k = 1.p1 (t) = p̂1 (θ) = 1,b1 = 1.2◦ . k = 2.p̂1 (θ) = θ + 1, b1 = 3/2,p̂2 (θ) = −θ,b2 = −1/2.3◦ . k = 3.231p̂1 (θ) = (θ + 1)(θ + 2), b1 = ,2124p̂2 (θ) = −θ(θ + 2),b2 = − ,3θ(θ + 1)5p̂3 (θ) =,b3 = .21217.1. ÌÅÒÎÄÛ ÀÄÀÌÑÀ175Âûïèøåì óðàâíåíèÿ (17.6) äëÿ ýòèõ ÷àñòíûõ ñëó÷àåâun+1 = un + τ f (un ),¸·13un+1 = un + τ f (un ) − f (un−1 ) ,22·¸23165un+1 = un + τf (un ) − f (un−1 ) + f (un−2 ) .121212(17.8)Óïðàæíåíèå 17.1.
Ïîñòðîèòü ÿâíûé 4-õ-øàãîâûé ìåòîä Àäàìñà (17.6).Îòâåò.·un+1¸5559379= un + τf (un ) − f (un−1 ) + f (un−2 ) − f (un−3 ) .24242424Çàìå÷àíèå 17.1. Î÷åâèäíî, ÷òî ïåðâîå èç óðàâíåíèé (17.8) îïðåäåëÿåò èññëåäîâàí-íûé íàìè ðàíåå ìåòîä Ýéëåðà. Òåì ñàìûì, ìåòîä Ýéëåðà ìîæåò áûòü îòíåñåí êàê êìåòîäàì Ðóíãå-Êóòòû, òàê è ê ìåòîäàì Àäàìñà.Ôîðìóëû (17.6) ïîëó÷åíû ïðè èíòåãðèðîâàíèè â ïðåäåëàõ îò tn äî tn+1 , â òîâðåìÿ êàê óçëû èíòåðïîëÿöèè ðàñïîëàãàëèñü íà îòðåçêå [tn+1−k , tn ], ò.å. âíå èíòåðâàëàèíòåãðèðîâàíèÿ (Äëÿ ïîäûíåãðàëüíîé ôóíêöèè èñïîëüçîâàëàñü ýêñòðàïîëÿöèÿ).
Âñâÿçè ñ ýòèì ÿâíûå ìåòîäû Àäàìñà èíîãäà íàçûâàþò ýêñòðàïîëÿöèîííûìè ìåòîäàìè.á) Íåÿâíûå ìåòîäû Àäàìñà. Ìîæíî ïîñòðîèòü è íåÿâíûå ìåòîäû Àäàìñà. Äëÿýòîãî ê óçëàì èíòåðïîëÿöèè (17.3) íóæíî äîáàâèòü åùå óçåë tn+1 .  ýòîì ñëó÷àåèíòåðïîëÿöèîííûé ìíîãî÷ëåí (ñòåïåíè k ) ïðèìåò âèäLk (t) =kXpj (t)f (u(tn+1−j )),(17.9)j=0à ñîîòâåòñòâóþùèì åìó óðàâíåíèåì áóäåò óðàâíåíèåun+1 − un = τkXbj f (un+1−j ),(17.10)j=0ãäå (ñð. ñ (17.7))1bj =τZZtn+11pj (t)dt =tn0kYθ−1+idθ.i−ji=0(17.11)i6=jÎïðåäåëåíèå 17.2. ×èñëåííûé ìåòîä (17.10), (17.11) íàçûâàåòñÿ íåÿâíûì k -øàãîâûììåòîäîì Àäàìñà (Èíîãäà åãî íàçûâàþò ìåòîäîì Àäàìñà-Ìóëòîíà).176 17. ËÈÍÅÉÍÛÅ ÌÍÎÃÎØÀÃÎÂÛÅ ÌÅÒÎÄÛÏðèìåðû.
4◦ . k = 0.p̂0 (θ) = 1,b0 = 1.5◦ . k = 1.p̂0 (θ) = θ,b0 = 1/2,p̂1 (θ) = −θ + 1, b1 = 1/2.6◦ . k = 2.15p̂0 (θ) = θ(θ + 1), b0 = ,21222p̂1 (θ) = −(θ − 1), b1 = ,311p̂2 (θ) = θ(θ − 1), b2 = − .212Íàïèøåì óðàâíåíèÿ (17.10) äëÿ ýòèõ ÷àñòíûõ ñëó÷àåâun+1 = un + τ f (un+1 ),τun+1 = un + [f (un+1 ) + f (un )] ,2·¸581un+1 = un + τf (un+1 ) + f (un ) − f (un−1 ) .121212(17.12)Óïðàæíåíèå 17.2. Ïîñòðîèòü íåÿâíûé 3-õ-øàãîâûé ìåòîä Àäàìñà (17.10), (17.11).Îòâåò.·un+1¸91951= un + τf (un+1 ) + f (un ) − f (un−1 ) + f (un−2 ) .24242424Çàìå÷àíèå 17.2. Î÷åâèäíî, ÷òî ïåðâîå èç óðàâíåíèé (17.12), îòâå÷àþùåå k = 0,ÿâëÿåòñÿ íåÿâíûì ìåòîäîì Ýéëåðà, à âòîðîå óðàâíåíèå, îòâå÷àþùåå k = 1, ìåòîäîìòðàïåöèé.
Òåì ñàìûì, ýòè îäíîøàãîâûå íåÿâíûå ìåòîäû Àäàìñà ÿâëÿþòñÿ è ìåòîäàìèÐóíãå-Êóòòû.17.2 Ôîðìóëû äèôôåðåíöèðîâàíèÿ íàçàäÂî âñåõ ïðåäûäóùèõ ñëó÷àÿõ, êàê ïðè ïîñòðîåíèè ìåòîäîâ Ðóíãå-Êóòòû, òàê è ïðèïîñòðîåíèè ìåòîäîâ Àäàìñà, ìû ïîëó÷àëè ÷èñëåííûå ìåòîäû ïóòåì èíòåãðèðîâàíèÿóðàâíåíèÿ (17.1) è çàìåíû ïîäûíåãðàëüíîé ôóíêöèè f (u) â (17.2) èíòåðïîëÿöèîííûììíîãî÷ëåíîì èëè çàìåíû èíòåãðàëà êâàäðàòóðíîé ôîðìóëîé. À ìîæíî ïîñòóïàòü èèíà÷å: èíòåðïîëÿöèîííûì ìíîãî÷ëåíîì çàìåíèòü u(t). Òîãäà äëÿ ïîñòðîåíèÿ ÷èñëåííîãî ìåòîäà íóæíî áóäåò âûðàæåíèå èíòåðïîëÿöèîííîãî ìíîãî÷ëåíà ïîäñòàâèòü â17.2. ÔÎÐÌÓËÛ ÄÈÔÔÅÐÅÍÖÈÐÎÂÀÍÈß ÍÀÇÀÄ177(17.1). ×òîáû ïîëó÷èëñÿ ÷èñëåííûé ìåòîä, òî÷êà tn+1 äîëæíà áûòü â ÷èñëå óçëîâèíòåðïîëÿöèè.
Ïóñòüu(t) ≈ Lk (t) =kXpj (t)u(tn+1−j ),(17.13)j=0ãäåpj (t) =kYi=0i6=jt − tn+1−i.tn+1−j − tn+1−iÏîäñòàâëÿÿ (17.13) â (17.1), ïîëó÷èì ïðèáëèæåííîå ðàâåíñòâîà k!kXXp0j (t)u(tn+1−j ) ≈ fpj (t)u(tn+1−j ) .j=0j=0Ïðåâðàòèì åãî â òî÷íîå ðàâåíñòâî â êàêîì-ëèáî óçëå.  ðåçóëüòàòå ïîëó÷èì óðàâíåíèå äëÿ îïðåäåëåíèÿ ïðèáëèæåííîãî ðåøåíèÿ. Ðàññìîòðèì ñëó÷àé, êîãäà óêàçàííûìóçëîì ÿâëÿåòñÿ tn+1 . Áóäåì èìåòükXp0j (tn+1 )un+1−j = f (un+1 ).j=0Êàê è ðàíüøå, ñäåëàåì ëîêàëüíóþ çàìåíó ïåðåìåííîé (t − tn )/τ = θ.
Òîãäàp0j (t) =dpj (t)1 dp̂j (θ)1== p̂0j (θ),dtτ dθτãäåpj (t) = p̂j (θ) =kYθ−1+i,i−ji=0i6=jè ïîëó÷åííûé ìåòîä ïðèíèìàåò âèäkXp̂0j (1)un+1−j = τ f (un+1 ).(17.14)j=0Îïðåäåëåíèå 17.3. ×èñëåííûå ìåòîäû (17.14) íàçûâàþòñÿ ôîðìóëàìè äèôôåðåí-öèðîâàíèÿ íàçàä.Ïðèìåðû. 7◦ . k = 1.p̂0 (θ) = θ,p̂00 (1) = 1,p̂1 (θ) = −θ + 1, p̂01 (1) = −1.178 17. ËÈÍÅÉÍÛÅ ÌÍÎÃÎØÀÃÎÂÛÅ ÌÅÒÎÄÛ8◦ . k = 2.13p̂0 (θ) = θ(θ + 1), p̂00 (1) = ,2220p̂1 (θ) = 1 − θ ,p̂1 (1) = −2,11p̂2 (θ) = θ(θ − 1), p̂02 (1) = .22Âûïèøåì óðàâíåíèÿ (17.14) äëÿ ýòèõ ñëó÷àåâun+1 − un = τ f (un+1 ),µ¶31un+1 − 2un + un−1 = τ f (un+1 ).22(17.15)(17.16)Óïðàæíåíèå 17.3.
Ïîñòðîèòü ôîðìóëó (17.14), îòâå÷àþùóþ k = 3.Îòâåò.µ1131un+1 − 3un + un−1 − un−2623¶= τ f (un+1 ).17.3 Îáùèå ëèíåéíûå ìíîãîøàãîâûå ìåòîäûÌåòîäû Àäàìñà, ÿâíûå è íåÿâíûå, è ôîðìóëû äèôôåðåíöèðîâàíèÿ íàçàä ÿâëÿþòñÿ÷àñòíûìè ñëó÷àÿìè ôîðìóëûkXαj un−j = τj=0kXβj f (un−j ),(17.17)j=0ãäå αj è βj äåéñòâèòåëüíûå ÷èñëà. (Îáðàòèì âíèìàíèå íà òî, ÷òî â ýòîé ôîðìóëåâìåñòî íîâîãî íåèçâåñòíîãî un+1 ôèãóðèðóåò un ). Áóäåò ïðåäïîëàãàòü, ÷òîα0 6= 0,|αk | + |βk | 6= 0.(17.18)Ïåðâîå èç óñëîâèé (17.18) îáåñïå÷èâàåò ðàçðåøèìîñòü íåÿâíîãî (β0 6= 0) óðàâíåíèÿ(17.17) ïî êðàéíåé ìåðå, äëÿ äîñòàòî÷íî ìàëîãî øàãà τ . Âòîðîå èç óñëîâèé (17.18)âñåãäà ìîæíî ñ÷èòàòü âûïîëíåííûì, óìåíüøèâ ïðè íåîáõîäèìîñòè k .Îïðåäåëåíèå 17.4.
Ôîðìóëà (17.17) íàçûâàåòñÿ ëèíåéíûì ìíîãîøàãîâûì ( k-øàãîâûì) ìåòîäîì.Ìåòîä ÿâëÿåòñÿ ÿâíûì, åñëè β0 = 0, è íåÿâíûì â ïðîòèâíîì ñëó÷àå.×òîáû ëèíåéûé ìíîãîøàãîâûé ìåòîä (17.17) ìîæíî áûëî èñïîëüçîâàòü äëÿ ÷èñëåííîãî ðåøåíèÿ çàäà÷è (17.1), íåîáõîäèìî, ÷òîáû óðàâíåíèå (17.17) àïïðîêñèìèðîâàëî óðàâíåíèå (17.1).17.3. ÎÁÙÈÅ ËÈÍÅÉÍÛÅ ÌÍÎÃÎØÀÃÎÂÛÅ ÌÅÒÎÄÛ179Îïðåäåëåíèå 17.5. Âåëè÷èíàkXk1Xψn =βj f (u(tn−j )) −αj u(tn−j )τ j=0j=0(17.19)íàçûâàåòñÿ ïîãðåøíîñòüþ àïïðîêñèìàöèè ìåòîäà (17.17).Âûÿñíèì âîïðîñ î ïîðÿäêå ïîãðåøíîñòè àïïðîêñèìàöèè ìåòîäà (17.17) ïðè τ → 0.Òåîðåìà 17.1.
Ìíîãîøàãîâûé ìåòîä (17.17) èìååò ïîãðåøùíîñòü àïïðîêñèìàöèèïîðÿäêà p 6 2R òîãäà è òîëüêî òîãäà, êîãäà âûïîëíÿþòñÿ ñëåäóþùèå óñëîâèÿkXkX¡αj = 0,j=0¢αj j q + qβj j q−1 = 0,q = 1, . . . , p.(17.20)j=0Äîêàçàòåëüñòâî. Ðàçëîæèì u(t) ïî ôîðìóëå Òåéëîðà â òî÷êå tn :u(t) =pX(t − tn )qq!q=0u(q) (tn ) + O((t − tn )p+1 ).(17.21)Òàê êàê f (u) = u0 (t), òî, äèôôåðåíöèðóÿ (17.21), ïîëó÷èìf (u(t)) =pX(t − tn )q−1 (q)qu (tn ) + O((t − tn )p ).q!q=0(17.22)Ïîäñòàâëÿÿ òåïåðü ðàçëîæåíèÿ (17.21), (17.22) ïðè t = tn−j â (17.19), áóäåì èìåòüψn =kXβjj=0pX(−jτ )q−1 (q)u (tn )−qq!q=0pk1 X X (−jτ )q (q)αju (tn ) + O(τ p ) =−τ j=0q!q=0=pX(−τ )q−1q=0q!u(q) (tn )kX£¤βj qj q−1 + αj j q + O(τ p ).j=0Ïðèðàâíèâàÿ íóëþ êîýôôèöèåíòûòû ïðè τ q−1 äëÿ q = 0, 1, . .
. , p, ïîëó÷èì (17.20).Òåîðåìà äîêàçàíà.Çàìå÷àíèå 17.3. Ðåøåíèå óðàâíåíèÿ (17.17) íå èçìåíèòñÿ, åñëè åãî óìíîæèòü íàêàêîå-ëèáî ÷èñëî, îòëè÷íîå îò íóëÿ. Ýòî îçíà÷àåò, ÷òî åãî êîýôôèöèåíòû îïðåäåëÿþòñÿ ñ òî÷íîñòüþ äî ìíîæèòåëÿ (äî ìóëüòèïëèêàòèâíîé ïîñòîÿííîé). ×òîáû óñòðàíèòüýòîò ïðîèçâîë, ïðîíîðìèðóåì èõ, ïîëàãàÿ, íàïðèìåð,kXj=0βj = 1.(17.23)180 17.
ËÈÍÅÉÍÛÅ ÌÍÎÃÎØÀÃÎÂÛÅ ÌÅÒÎÄÛÇàìå÷àíèå 17.4. Èç (17.20), (17.23) èìååì (p + 2) óðàâíåíèÿ äëÿ 2(k + 1) êîýôôèöè-åíòîâ ìåòîäà (17.17). Òåì ñàìûì, ìàêñèìàëüíûé ïîðÿäîê àïïðîêñèìàöèè ëèíåéíîãîk -øàãîâîãî ìåòîäà åñòü p = 2k .17.4 Ïîãðåøíîñòü àïïðîêñèìàöèè ìåòîäîâ ÀäàìñàÈññëåäóåì âîïðîñ î ïîðÿäêå ïîãðåøíîñòè àïïðîêñèìàöèè ìåòîäîâ Àäàìñà. Äëÿ ýòîãîïåðåïèøåì ñíà÷àëà ÿâíûé ìåòîä Àäàìñà (17.6), (17.7) â âèäå (17.17), ò.å.
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