Численные методы. Ионкин (2009) (1160433), страница 8
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. , xm ) = 0,f (x , . . . , x ) = 0,2 1m...fm (x1 , . . . , xm ) = 0,òî òàêæå ìîæíî èñïîëüçîâàòü ìåòîä Íüþòîíà, â ýòîì ñëó÷àåI(xn )ij =∂fi (xn ),∂xji, j = 1, mÑèñòåìà â ýòîì ñëó÷àå èìååò òîò æå âèä:f (xn ) + I(xn )(xn+1 − xn ) = 0Ìåòîä ñåêóùèõÇàïèøåì ìåòîä Íüþòîíà:xn+1 = xn −Çàìåíèì â íåìf 0 (xn )íàf (xn ),f 0 (xn )x0 ∈ Ua (x∗ ),n = 0, 1, 2, . . . .f (xn )−f (xn−1 ).xn −xn−1Ïîëó÷èìxn+1xn − xn−1=x −f (xn )nn−1f (x ) − f (x )nÏîñêîëüêó â çàïèñè äàííîãî ìåòîäà ó÷àâñòâóþò òðè ïîñëåäîâàòåëüíûå èòåðàöèè (x(6)n+1, xn è xn−1 ),òî îí íàçûâàåòñÿ äâóõøàãîâûì ìåòîäîì.
Äëÿ òîãî, ÷òîáû âîñïîëüçîâàòüñÿ èì, òðåáóåòñÿ çàäàòü01äâà íà÷àëüíûõ ïðèáëèæåíèÿ (x è x ). Èõ ìîæíî ïîëó÷èòü ìåòîäîì ïðîñòîé èòåðàöèè èëèìåòîäîì Íüþòîíà.Çàìåòèì, ÷òî, èñïîëüçóÿ ìåòîä ñåêóùèõ, ìû ïîëó÷àåìxn+1ïðè ïîìîùè èíòåðïëÿöèè ôóíêxn èöèè f ïîëèíîìîì ïåðâîé ñòåïåíè (ëèíåéíîé ôóíêöèåé), èñïîëüçóÿ åå çíà÷åíèå â óçëàõxn−1 .Ñõîäèìîñòü ìåòîäà Íüþòîíà è îöåíêà ñõîäèìîñòè458Ñõîäèìîñòü ìåòîäà Íüþòîíà è îöåíêà ñõîäèìîñòèÐàññàìòðèâàåòñÿ íåëèíåéíîå óðàâíåíèåf (x) = 0.(1)Çàïèøåì äëÿ íåãî ìåòîä Íüþòîíà:xn+1 = xn −f (xn ),f 0 (xn )n = 0, 1, . .
. ;x0 ∈ Ua (x∗ ).(2)Çàïèøåì ýòî ìåòîä â áîëåå îáùåì âèäå:xn+1 = S(xn ),ÒîãäàãäåS(x) = x −f (x).f 0 (x)(f 0 (x))2 − f (x)f 00 (x)f (x)f 00 (x)S (x) = 1 −=.(f 0 (x))2(f 0 (x))20S 0 (x∗ ) = 0.zn = xn − x∗ ïîãðåøíîñòü.Çàìåòèì, ÷òîÏóñòüÒîãäàz n+1 = xn+1 − x∗ = S(xn ) − S(x∗ ) = S(zn + x∗ ) − S(x∗ ).Âîñïîëüçóåìñÿ ôîðìóëîé Òåéëîðà ñ îñòàòî÷íûì ÷ëåíîì â ôîðìå Ëàãðàíæà:11z n+1 = S(x∗ ) + S 0 (x∗ )zn + S 00 (x̃n )zn2 − S(x∗ ) = S 00 (x̃n )zn2 ,22ãäåx̃n = xn + θzn , |θ| < 1.Ïóñòü ∃M > 0 òàêîå, ÷òî1 00|S (x)| ≤ M,2x ∈ Ua (x∗ ).(3)Òîãäà|zn+1 | ≤ M |zn |2 ,M |zn+1 | ≤ (M |zn |)2 .Ïðèìåíèì ýòî íåðàâåíñòâî ðåêóðñèâíî, ïîëó÷èìnM |zn | ≤ (M |z0 |)2 ,1n(M |z0 |)2 .M|zn | → 0 ⇒ xn → x∗ .|zn | ≤ÅñëèM |z0 | < 1,òî ïðèn→∞ïîëó÷àåìÒàêèì îáðàçîì, äëÿ ñõîäèìîñòè äàííîãî ìåòîäà äîñòàòî÷íî ïîòðåáîâàòü1.M(4)1n(M |x0 − x∗ |)2 .M(5)|z0 | = |x0 − x∗ | ≤Äëÿznèìååì îöåíêó|zn | = |xn − x∗ | ≤Ìû äîêàçàëè ñëåäóþùóþ òåîðåìó.Ñõîäèìîñòü ìåòîäà Íüþòîíà è îöåíêà ñõîäèìîñòèÒåîðåìà∃M > 0(îáîöåíêåòàêîå, ÷òîñêîðîñòè59ñõîäèìîñòè0 1 f (x)f 0 (x) ≤M2 (f 0 (x))2 |x0 − x∗ | ≤ìåòîäàÍüþòîíà).Ïóñòü∀x ∈ Ua (x∗ ),1.MÒîãäà ìåòîä Íüþòîíà ñõîäèòñÿ è èìååò ìåñòî îöåíêà|xn − x∗ | ≤1n(M |x0 − x∗ |)2 .MÇàìå÷àíèå.
Åñëè ìåòîä Íüþòîíà ñõîäèòñÿ, òî îí ñõîäèòñÿ î÷åíü áûñòðî.Çàìå÷àíèå. Íà÷àëüíîå ïðèáëèæåíèå äîëæíî áûòü áëèçêî ê êîðíþ (â ñîîòâåòñòâèè ñ óñëîâèåì (4)).Íàïîìíèì, ÷òî ìîäèôèöèðîâàííûé ìåòîä Íüþòîíà èìååò âèä:Äëÿ ýòîãî ìåòîäàS(x)xn+1 = xn −f (xn ).f 0 (x0 )S(x) = x −f (x).f 0 (x0 )èìååò âèäÄëÿ ýòîãî ìåòîäà àíàëîãè÷íîå óòâåðæäåíèå íå èìååò ìåñòî, èáîS 0 (x∗ ) 6= 0â îáùåì ñëó÷àå.Ãëàâà IVÐàçíîñòíûå ìåòîäû ðåøåíèÿ çàäà÷ìàòåìàòè÷åñêîé ôèçèêè1Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòèÐàññìîòðèì îáëàñòüD = {(x, y) ∈ R2 : 0 < x < 1, 0 < t ≤ T } (T çàäàííîå ïîëîæèòåëüíîå÷èñëî).Çàïèøåì ïåðâóþ êðàåâóþ çàäà÷ó äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòè â ýòîé îáëàñòè:∂ 2u∂u= 2 + f (x, t), (x, t) ∈ D,∂t∂t(u(0, t) = µ1 (t),u(1, t) = µ2 (t),êðàåâûå óñëîâèÿ:(1)(2)íà÷àëüíîå óñëîâèå:u(x, 0) = u0 (x).(3)Ââåäåì ñëåäóþùèå îáîçíà÷åíèÿ:ωh = {xi = ih, i = 1, .
. . , N − 1, hN = 1},ω h = {xi = ih, i = 0, . . . , N, hN = 1},ωτ = {tj = jτ, j = 1, . . . , j0 , τ j0 = T },ω τ = {tj = jτ, j = 0, . . . , j0 , τ j0 = T },ωτ h = ωτ × ωh ,ωτ h = ωτ × ωh,uni = u(xi , tn ),fin = f (xi , tn ).Ìíîæåñòâàω∗èω∗íàçûâàþòñÿ ñåòêàìè, ýëåìåíòû ýòèõ ìíîæåñòâ óçëàìè. Çíà÷åíèÿíàçûâàþòñÿ øàãàìè ñåòêè. Âíóòðåííèìè óçëàìè íàçîâåì óçëû ñåòêèÁóäåì îáîçíà÷àòü ÷èñëåííîå ðåøåíèå ïîñòàâëåííîé çàäà÷è ÷åðåçyin = y(xi , tn ).60ωτ h .y(x, t).ÏóñòüτèhÐàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòè61ßâíàÿ ðàçíîñòíàÿ ñõåìàÇàïèøåì ðàññìàòðèâàåìóþ çàäà÷ó:∂u∂ 2u= 2 + f (x, t), 0 < x < 1,∂t∂t(u(0, t) = µ1 (t),u(1, t) = µ2 (t),êðàåâûå óñëîâèÿ:0 < t ≤ T,(4)(5)íà÷àëüíîå óñëîâèå:u(x, 0) = u0 (x).(6)Ðàçíîñòíûé àíàëîã çàäà÷è (4) (6) èìååò âèä:ny n − 2yin + yi+1yin+1 − yin= i−1+ f (xi , tn ), (xi , tn ) ∈ ωτ h ,τh2(y0n+1 = µ1 (tn+1 ), tn+1 ∈ ω τ ,n+1yN= µ2 (tn+1 ), tn+1 ∈ ω τ ,yi0 = u0 (xi ),Ìíîæåñòâî óçëîâ{(xi , tn ), i = 0, .
. . , N }(7)(8)xi ∈ ω h .íàçûâàåòñÿn-ì(9)ñëîåì.Ïðè èçó÷åíèè ðàçíîñòíûõ ñõåì âîçíèêàþò ñëåäóþùèå âîïðîñû:1. Ñóùåñòâîâàíèå è åäèíñòâåííîñòü ðåøåíèÿ2. Ïîãðåøíîñòü àïïðîêñèìàöèè ðàçíîñòíîé ñõåìû3. Àëãîðèòì íàõîæäåíèÿ ÷èñëåííîãî ðåøåíèÿ4. Èññëåäîâàíèå óñòîé÷èâîñòè ðàçíîñòíîé ñõåìû5. Îöåíêà ñêîðîñòè ñõîäèìîñòè ðàçíîñòíîé ñõåìûÎòâåòèì íà âîïðîñû 1 è 3 äëÿ ÿâíîé ðàçíîñòíîé ñõåìû.
Ïåðåïèøåì (7) â âèäåτ nn(y − 2yin + yi+1) + τ fin , i = 1, . . . , N − 1.h2 i−1óçëàõ (i = 0, i = N ) çàäàíû ôîðìóëàìè (8). Çíà÷åíèÿ yyin+1 = yin +Çíà÷åíèÿyâ ãðàíè÷íûõ(10)ïðèn=0 ôîðìóëîé (9). Òàêèì îáðàçîì, ðåøåíèå ÿâíîé ðàçíîñòíîé ñõåìû ñóùåñòâóåò è åäèíñòâåííî èâûïèñàí àëãîðèòì åãî íàõîæäåíèÿ. Çàäà÷à ðåøàåòñÿ ïî ñëîÿì, ò.å. çíà÷åíèÿ íàíàõîäÿòñÿ ïî ÿâíîé ôîðìóëå ïî èçâåñòíûì çíà÷åíèÿì íànÎïðåäåëèì ïîãðåøíîñòü ðàçíîñòíîé ñõåìû xi òàê:n-ì(n + 1)-ìñëîåñëîå.xni = yin − uni .Ââåäåì ôóíêöèþψinòàê:ψinuni−1 − 2uni + uni+1 un+1− unii=−+ fin .2hτ(11)Òîãäà (7) ìîæíî ïåðåïèñàòü ñëåäóþùèì îáðàçîì:nz n − 2zin + zi+1zin+1 − zin= i−1+ ψin ,τh2(xi , tn ) ∈ ωτ h .(12)Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòèÎïðåäåëåíèå. Ôóíêöèÿ ψin , îïðåäåëÿåìàÿ ðàâåíñòâîì62(11), íàçûâàåòñÿ ïîãðåøíîñòüþ àï-ïðîêñèìàöèè ðàçíîñòíîé ñõåìû (7) (9) íà ðåøåíèå çàäà÷è (4) (6).Çàäà÷à.
Äîêàçàòü, ÷òî ψin = O(τ + h2 ).Ðåøåíèå. Ðàçëîæèìu(xi , tn+1 )â óçëå(xi , tn )ïî ôîðìóëå Òåéëîðà:u(xi , tn+1 ) = un+1= u(xi , tn ) + ut (xi , tn )τ + O(τ 2 ).iÐàçëîæèìu(xi+1 , tn )â óçëå(xi , tn )ïî ôîðìóëå Òåéëîðà:11u(xi+1 , tn ) = uni+1 = u(xi , tn ) + ux (xi , tn )h + uxx (xi , tn )h2 + uxxx (xi , tn )h3 + O(h4 ).26Ðàçëîæèìu(xi−1 , tn )â óçëå(xi , tn )ïî ôîðìóëå Òåéëîðà:11u(xi−1 , tn ) = uni+1 = u(xi , tn ) − ux (xi , tn )h + uxx (xi , tn )h2 − uxxx (xi , tn )h3 + O(h4 ).26Ïîäñòàâèâ âûïèñàííûå ðàçëîæåíèÿ â (11), ïðèâåäÿ ïîäîáíûå ÷ëåíû è âîñïîëüçîâàâøèñü (4),ïîëó÷èìψin = O(τ + h2 ).Êðàåâûå óñëîâèÿ äëÿzèìåþò âèä:n+1z0n+1 = zN= 0,À íà÷àëüíîå óñëîâèå äëÿtn+1 = ω τ .(13)z:zi0 = 0,xi = ω h .(14)Ââåäåì íîðìó íà ñëîå:ky n kC = max |yin |.0≤i≤NÂâåäåííàÿ òàêèì îáðàçîì íîðìà íàçûâàåòñÿ ðàâíîìåðíîé (ñèëüíîé).n+1Âûðàçèì ziâ ôîðìóëå (12):zin+1 = zin +τ nn) + τ ψin .(z − 2zin + zi+1h2 i−1Ïîòðåáóåì âûïîëíåíèÿ ñëåäóþùåãî óñëîâèÿ:τ1=γ≤ .2h2(15)Åñëè ðàçíîñòíàÿ ñõåìà ñõîäèòñÿ ïðè îãðàíè÷åíèè íà øàãè ñåòêè, òî òàêàÿ ðàçíîñòíàÿ ñõåìàíàçûâàåòñÿ óñëîâíî ñõîäÿùåéñÿ.
Åñëè ñõîäèìîñòü ðàçíîñòíîé ñõåìû íå çàâèñèò îò øàãîâ ñåòêè,òî ðàçíîñòíàÿ ñõåìà íàçûâàåòñÿ àáñîëþòíî ñõîäÿùåéñÿ.Äîêàæåì, ÷òî óñëîâèå (15) ÿâëÿåòñÿ íåîáõîäèìûì è äîñòàòî÷íûì äëÿ ñõîäèìîñòè (è óñòîé÷èâîñòè) ÿâíîé ðàçíîñòíîé ñõåìû.Äîêàæåì äîñòàòî÷íîñòü óñëîâèÿ (15). Ïóñòü ýòî óñëîâèå âûïîëíåíî. Òîãäànnzin+1 = (1 − 2γ)zin + γ(zi−1+ zi+1) + τ ψin ,Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòè63nn|) + τ ψin ,| + |zi+1|zin+1 | ≤ (1 − 2γ)|zin | + γ(|zi−1|zin+1 | ≤ (1 − 2γ)kz n kC + γ(kz n kc + kz n kC ) + τ kψ n kC ,|zin+1 | ≤ kz n kC + τ kψ n kC ,ïîñêîëüêó ýòî âûïîëíÿåòñÿ äëÿ âñåõi,òîkz n+1 kC ≤ kz n kC + τ kψ n kC .(16)Ïðèìåíÿÿ ôîðìóëó (16) êàê ðåêóððåíòíóþ, ïîëó÷èìkzn+10kC ≤ kz kC + τnXkψ k kC ,k=0kz n+1 kC = 0,ïîñêîëüêóòîkz n+1 kC ≤ τnXkψ k kC .k=02= O(τ + h ), òî ∃ M > 0 : kψ kC ≤ M (τ + h2 ), MnPτ = tn+1 ≤ T, èìååìÓ÷èòûâàÿ, ÷òînÒ.ê.
ψiníå çàâèñèò îòτèh.k=0kz n+1 kC ≤ M T (τ + h2 ) = M1 (τ + h2 ).Ïðè ýòîì,M1íå çàâèñèò îòτèh.Ìû ïîëó÷èëè àïðèîðíóþ îöåíêókz n+1 kC ≤ M1 (τ + h2 ).(17)Èç ïîëó÷åííîé îöåíêè ñëåäóåò, ÷òîτ, h → 0 ⇒ kz n+1 k → 0,ò.å.ky n+1 − un−1 k → 0.Òàêèì îáðàçîì, èìååò ìåñòî ñõîäèìîñòü ÷èëñåííîãî ðåøåíèÿ ê ðåøåíèþ èñõîäíîé çàäà÷è.Íåñêîëüêî ñëîâ îá óñòîé÷èâîñòè.Ïóñòüy(0, t) = y(1, t) = 0.Òîãäà, ïðîâåäÿ ðàññóæäåíèÿ, àíàëîãè÷íûì îïèñàííûì âûøå,èìååìkyn+1kC ≤ ky0 kC +nXτ kf k kC ,k=0kyn+1kC ≤ ky0 kC + τnXkf k kC .(18)k=0Ðàçíîñòíóþ ñõåìó, â êîòîðîé âûïîëíÿåòñÿ (18), íàçûâàþò óñòîé÷èâîé ïî íà÷àëüíîìó óñëîâèþ è ïðàâîé ÷àñòè. Òàêèì îáðàçîì, ÿâíàÿ ðàçíîñòíàÿ ñõåìà óñòîé÷èâà ïî íà÷àëüíîìó óñëîâèþè ïðàâîé ÷àñòè ïðè âûïîëíåíèè óñëîâèÿ (15).Äîêàæåì, ÷òî óñëîâèå (15) ÿâëÿåòñÿ íåîáõîäèìûì äëÿ ñõîäèìîñòè ÿâíîé ðàçíîñòíîé ñõåìû.Ðàññìîòðèì îäíîðîäíóþ ñèñòåìónnyi−1− 2yin + yi+1yin+1 − yin,=τh2(xi , tn ) ∈ ωτ h .(19)Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòèÁóäåì èñêàòü åå ðåøåíèå â âèäåyjn = q n eijhφ ,ãäåi2 = −1, φ ∈ R, q ∈ C.64Ïîäñòàâèì ýòî âóðàâíåíèå (19).
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