Н.И. Ионкин - Электронные лекции (2009) (1135239), страница 9
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Òàêèì îáðàçîì, ðåøåíèåÿâíîé ðàçíîñòíîé ñõåìû ñóùåñòâóåò è åäèíñòâåííî è âûïèñàí àëãîðèòìåãî íàõîæäåíèÿ. Çàäà÷à ðåøàåòñÿ ïî ñëîÿì, ò.å. çíà÷åíèÿ íàñëîå íàõîäÿòñÿ ïî ÿâíîé ôîðìóëå ïî èçâåñòíûì çíà÷åíèÿì íànÎïðåäåëèì ïîãðåøíîñòü ðàçíîñòíîé ñõåìû xi òàê:(n + 1)-ìn-ì ñëîå.xni = yin − uni .Ââåäåì ôóíêöèþψinòàê:uni−1 − 2uni + uni+1 un+1− unii−+ fin .h2τψin =(11)Òîãäà (7) ìîæíî ïåðåïèñàòü ñëåäóþùèì îáðàçîì:nz n − 2zin + zi+1zin+1 − zin= i−1+ ψin ,τh2(xi , tn ) ∈ ωτ h .Îïðåäåëåíèå. Ôóíêöèÿ ψin , îïðåäåëÿåìàÿ ðàâåíñòâîì(12)(11), íàçûâàåò-ñÿ ïîãðåøíîñòüþ àïïðîêñèìàöèè 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 ).iÐàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿòåïëîïðîâîäíîñòèÐàçëîæèì82u(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 ÷ëåíûè 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 .(zi−1 − 2zin + zi+12hÏîòðåáóåì âûïîëíåíèÿ ñëåäóþùåãî óñëîâèÿ:1τ=γ≤ .2h2(15)Åñëè ðàçíîñòíàÿ ñõåìà ñõîäèòñÿ ïðè îãðàíè÷åíèè íà øàãè ñåòêè, òîòàêàÿ ðàçíîñòíàÿ ñõåìà íàçûâàåòñÿ óñëîâíî ñõîäÿùåéñÿ.
Åñëè ñõîäèìîñòü ðàçíîñòíîé ñõåìû íå çàâèñèò îò øàãîâ ñåòêè, òî ðàçíîñòíàÿ ñõåìàíàçûâàåòñÿ àáñîëþòíî ñõîäÿùåéñÿ.Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿòåïëîïðîâîäíîñòè83Äîêàæåì, ÷òî óñëîâèå (15) ÿâëÿåòñÿ íåîáõîäèìûì è äîñòàòî÷íûì äëÿñõîäèìîñòè (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 ,nn|zin+1 | ≤ (1 − 2γ)|zin | + γ(|zi−1| + |zi+1|) + τ ψin ,|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=0ïîñêîëüêókz n+1 kC = 0,òîkzn+1kC ≤ τnXkψ k kC .k=0ψin = O(τ + h2 ), òî ∃ M > 0 : kψ n kC ≤ M (τ + h2 ), Mτ è h.nPÓ÷èòûâàÿ, ÷òîτ = tn+1 ≤ T, èìååìÒ.ê.îòíå çàâèñèòk=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 ).Èç ïîëó÷åííîé îöåíêè ñëåäóåò, ÷òîτ, h → 0 ⇒ kz n+1 k → 0,ò.å.ky n+1 − un−1 k → 0.(17)Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿòåïëîïðîâîäíîñòè84Òàêèì 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÷íûìîïèñàííûì âûøå, èìååìky n+1 kC ≤ ky0 kC +nXτ kf k kC ,k=0ky n+1 kC ≤ 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 óñëîâèþ è ïðàâîé ÷àñòè ïðèâûïîëíåíèè óñëîâèÿ (15).Äîêàæåì, ÷òî óñëîâèå (15) ÿâëÿåòñÿ íåîáõîäèìûì äëÿ ñõîäèìîñòèÿâíîé ðàçíîñòíîé ñõåìû. Ðàññìîòðèì îäíîðîäíóþ ñèñòåìóÁóäåìC.ny n − 2yin + yi+1yin+1 − yin= i−1, (xi , tn ) ∈ ωτ h .(19)τh2nn ijhφèñêàòü åå ðåøåíèå â âèäå yj = q e, ãäå i2 = −1, φ ∈ R, q ∈Ïîäñòàâèì ýòî â óðàâíåíèå (19).
Ïîëó÷èìq = 1 + γ(eihφ − 2 + e−ihφ ) = 1 + γ(2 cos hφ − 2) = 1 − 4γ sin2Åñëè âçÿòühφ.2φ òàêîå, ÷òî |q| > 1, ò.å. γ > 12 , òî ãàðìîíèêè áóäóò íåîãðà-íè÷åííî âîçðàñòàòü è ðàçíîñòíàÿ ñõåìà áóäåò ðàñõîäèòüñÿ.Òàêèì îáðàçîì, óñëîâèå (15) ÿâëÿåòñÿ íåîáõîäèìûì è äîñòàòî÷íûìäëÿ ñõîäèìîñòè è óñòîé÷èâîñòè ÿâíîé ðàçíîñòíîé ñõåìû.×èñòî íåÿâíàÿ ðàçíîñòíàÿ ñõåìà (ñõåìà ñ îïåðåæåíèåì)Çàïèøåì ðàññìàòðèâàåìóþ çàäà÷ó:∂u∂ 2u= 2 + f (x, t),∂t∂t0 < x < 1,0 < t ≤ T,(20)Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿòåïëîïðîâîäíîñòè85êðàåâûå óñëîâèÿ:(u(0, t) = µ1 (t),u(1, t) = µ2 (t),(21)íà÷àëüíîå óñëîâèå:u(x, 0) = u0 (x).(22)Ðàçíîñòíûé àíàëîã çàäà÷è (20) (22) èìååò âèä:n+1n+1yi−1− 2yin+1 + yi+1yin+1 − yin=+ f (xi , tn+1 ),τh2(y0n+1 = µ1 (tn+1 ),n+1yN= µ2 (tn+1 ),yi0 = u0 (xi ),(xi , tn+1 ) ∈ ωτ h ,tn+1 ∈ ω τ ,tn+1 ∈ ω τ ,(23)(24)xi ∈ ω h .(25)Ïåðåïèøåì (23) â âèäå:n+1n+1γyi−1− (1 + 2γ)yin+1 + γyi+1= −(yin + fin+1 ),i = 1, .
. . , N − 1.Äàííàÿ ñèñòåìà óðàâíåíèé ñîñòîèò èç òðåõòî÷å÷íûõ óðàâíåíèé. ÅåìàòðèöàAÿâëÿåòñÿ òðåõäèàãîíàëüíîé. Ýòà ñèñòåìà ðåøàåòñÿ ìåòîäîìïðîãîíêè. Ìîæíî äîêàçàòü, ÷òî|A| =6 0. Òàêèì îáðàçîì, ðåøåíèå äàííîéñèñòåìû ñóùåñòâóåò è åäèíñòâåííî, è íàõîäèòñÿ ìåòîäîì ïðîãîíêè.Ââåäåì ïîãðåøíîñòü:zin = yin − u(xi , tn ) = yin − uniÒîãäà äëÿ ïîãðåøíîñòè ïîëó÷èì óðàâíåíèå:n+1z n+1 − 2zin+1 + zi−1zin+1 − zin+ ψin ,= i+12τhãäåψin =n+1un+1+ un+1un+1− unii+1 − 2uii−1i−+ fin+12hτÇàäà÷à.
Ïîêàçàòü, ÷òî ψin èç(27) åñòüO(τ + h2 ).(26)(27)Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿòåïëîïðîâîäíîñòè86un+1i±1Ðåøåíèå. Ðàçëîæèìèuniâ ðÿä Òåéëîðà:n+1n+1un+1± un+1i±1 = uix,i h + uxx,ih2h3± un+1+ O(h4 )xxx,i262uni = un+1− un+1it,i τ + O(τ )Ïîäñòàâèì ýòè ðàçëîæåíèÿ â ôîðìóëó (27). Ïîëó÷èì:n+1n+1ψin = (−un+1) + O(τ + h2 ) = O(τ + h2 )t,i + uxx,i + fiÇàìåòèì, ÷òî:n+1z0n+1 = zN= zi0 = 0,Ïóñòü∃i0 ,i = 0, . . .
, N(28)òàêîé ÷òî:| = max |zin+1 | = ||z n+1 ||C|zin+101≤i≤Nn+1n+1zin+1 = zin + γ(zi+1− 2zin+1 + zi−1) + τ ψin ,γ=τh2n+1n+1(1 + 2γ)zin+1 = zin + γ(zi+1+ zi−1) + τ ψinÇàïèøåì ïîñëåäíåå ðàâåíñòâî äëÿ óçëài0 :(1 + 2γ)zin+1= zin0 + γ(zin+1+ zin+1) + τ ψin000 +10 −1(1 + 2γ)|zin+1| ≤ |zin0 | + γ(|zin+1| + |zin+1|) + τ |ψin0 |00 +10 −1(1 + 2γ)||z n+1 ||C ≤ ||z n ||C + 2γ||z n+1 ||C + τ ||ψ n ||C||z n+1 ||C ≤ ||z n ||C + τ ||ψ n ||CÏîñëåäíåå ñîîòíîøåíèå ÿâëÿåòñÿ ðåêêóðåíòíûì.
Ïðèìåíèì åãî n ðàç:||zn+10||C ≤ ||z ||C +NXτ ||ψ k ||Ck=0Èç (28) èìååì:îòτ||z 0 ||C = 0.Òàê êàê||ψ k || ≤ M (τ + h2 ),è h, òî:||z n+1 ||C ≤ MNXk=0τ (τ + h2 )ãäå M íå çàâèñèòÐàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿòåïëîïðîâîäíîñòè87Òàêèì îáðàçîì, îêîí÷àòåëüíî ïîëó÷àåì:||z n+1 ||C ≤ M1 (τ + h2 ),M1 = M tn+1 íå çàâèñèò îòτè h.Èç ïîñëåäíåãî ñîîòíîøåíèÿ ñëåäóåò, ÷òî ÷èñòî íåÿâíàÿ ðàçíîñòíàÿ ñõåìààñáîëþòíî ñõîäèòñÿ (èìååì àáñîëþòíóþ ñõîäèìîñòü ïåðâîãî ïîðÿäêà ïîn+1τ è âòîðîãî ïîðÿäêà ïî h). Åñëè y0n+1 = yN= 0, òî:||yn+1||C ≤ ||u0 ||C +NXτ ||f k ||Ck=0Òàêèì îáðàçîì, ïîëó÷àåì óñòîé÷èâîñòü ÷èñòî íåÿâíîé ðàçíîñòíîé ñõåìûïî íà÷àëüíîìó ïðèáëèæåíèþ è ïðàâîé ÷àñòè.Ñèììåòðè÷íàÿ ðàçíîñòíàÿ ñõåìà (ñõåìà Êðàíêà-Íèêîëüñîíà)Îáîçíà÷èì ÷åðåçmyxx,iâòîðóþ ðàçíîñòíóþ ïðîèçâîäíóþ ïî ïðîñòðàí-ñòâåííîé ïåðåìåííîé:m=yxx,immyi+1− 2yim − yi−1h2Ðàçíîñòíàÿ ñõåìà èìååò âèä:yin+1 − yinn+1n= 0.5(yxx,i+ yxx,i) + f (xi , tn + 0.5τ )τn+1y0n+1 = µ1 (tn+1 ), yN= µ2 (tn+1 ),yi0 = u0 (xi ),Ââåäåì ïîãðåøíîñòü:zin = yin − uni .xi ∈ ω hn+1ψin = 0.5(uxx,i+ unxx,i ) −Çàäà÷à.
Ïîêàçàòü, ÷òî ψin èç(30)(31)Òîãäà äëÿ ïîãðåøíîñòè èìååì:zin+1 − zinn+1n= 0.5(zxx,i+ zxx,i) + ψin ,τn+1= 0,z0n+1 = zNtn+1 ∈ ω t(29)zi0 = 0,(xi , tn+1 ) ∈ ωτ hi = 0, . . . , Nun+1− unii+ f (xi , tn + 0.5τ )τ(34) åñòüO(τ 2 + h2 ).(32)(33)(34)Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿòåïëîïðîâîäíîñòè88Ðåøåíèå. Ðàçëîæèìnun+1i±1 è ui â ðÿä Òåéëîðà â îêðåñòíîñòè òî÷êè (xi , tn+ 1 ):21 n+ 1 τ 2τ+ utt,i 2+ O(τ 3 )2 221 n+ 1 τ 2n+ 1 τ+ O(τ 3 )− ut,i 2 + utt,i 22 22n+ 21un+1= uiin+ 12uni = uin+ 12+ ut,iÏîäñòàâèì ýòè ðàçëîæåíèÿ â ôîðìóëó (34):n+ 12ψin = −ut,in+ 12n+ O(τ 2 ) + 0.5(un+1xx,i + uxx,i ) + fiÒåïåðü â ïðåäñòàâëåíèè âòîðîé ðàçíîñòíîé ïðîèçâîäíîé ðàçëîæèì âñåâõîæäåíèÿ ôóíêöèè â ðÿä Òåéëîðà. Ïðèâîäÿ ïîäîáíûå ñëàãàåìûå, ïîëó÷èì:unxx,i = unxx,i + unxxxx,ih2+ O(h4 )12un+1xx,i , à çàòåì ïðîâåäåì åùå îäíî ðàçëîæåíèå(xi , tn+ 1 ):Ïðèìåíèì ýòî ðàçëîæåíèå êâ ðÿä Òåéëîðà â òî÷êåun+1xx,i2=un+1xx,i+2n+1 huxxxx,i12+ O(h4 ) =2τh2 τn+ 12 hn+ 12+ uxxxx,i+ uxxxxt,i· + O(τ 2 + h4 )21212 2nïðîäåëàåì è ñ uxx,i :n+ 1n+ 1= uxx,i2 + uxxt,i2Òî æå ñàìîåunxx,i = unxx,i + unxxxx,ih2+ O(h4 ) =122τh2 τn+ 12 hn+ 12+ uxxxx,i− uxxxxt,i· + O(τ 2 + h4 )21212 2nðàçëîæåíèÿ â âûðàæåíèå äëÿ ψi è ó÷òåì óðàâíåíèån+ 1n+ 1= uxx,i2 − uxxt,i2Ïîäñòàâèì ýòèòåï-ëîïðîâîäíîñòè:n+ 12ψin = (−ut,in+ 1n+ 12+ uxx,i2 + fin+ 12) + uxxxx,ih2+ O(τ 2 + h4 ) = O(τ 2 + h2 )12Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿòåïëîïðîâîäíîñòè89Çàäà÷à Øòóðìà-ËèóâèëëÿÐàññìîòðèì çàäà÷ó Øòóðìà-Ëèóâèëëÿ äëÿ äèôôåðåíöèàëüíîãî óðàâíåíèÿ âòîðîãî ïîðÿäêà:u00 (x) + λu(x) = 0,u(0) = u(1) = 0;0 < x < 1,u(x), íå ðàâíûå òîæäåñòâåííî íóëþ, - ñîáñòâåííûå ôóíêöèè ÇØË, àλ- ñîáñòâåííûå çíà÷åíèÿ ÇØË.
Ðåøåíèåì äàííîé çàäà÷è ÿâëÿþòñÿ ñîáñòâåííûå çíà÷åíèÿλkè ñîáñòâåííûå ôóíêöèèλk = (πk)2 ,uk (x):k = 1, 2, . . .0 < λ1 < λ2 < · · · < λn < . . .uk (x) = C sin(πkx)Ðàññìîòðèì ëèíåéíîå ïðîñòðàíñòâî L2 ôóíêöèé, èíòåãðèðóåìûõ ñ êâàäR1 2ðàòîì íà [0, 1] ( f (x)dx < ∞).
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