Численные методы. Ионкин (2009), страница 9

Ïîëó÷èìq = 1 + γ(eihφ − 2 + e−ihφ ) = 1 + γ(2 cos hφ − 2) = 1 − 4γ sin2Åñëè âçÿòüφòàêîå, ÷òî|q| > 1,ò.å.γ > 21 ,hφ.2òî ãàðìîíèêè áóäóò íåîãðàíè÷åííî âîçðàñòàòü èðàçíîñòíàÿ ñõåìà áóäåò ðàñõîäèòüñÿ.Òàêèì îáðàçîì, óñëîâèå (15) ÿâëÿåòñÿ íåîáõîäèìûì è äîñòàòî÷íûì äëÿ 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)êðàåâûå 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.A ÿâëÿåòñÿ òðåõ÷òî |A| =6 0. ÒàêèìÄàííàÿ ñèñòåìà óðàâíåíèé ñîñòîèò èç òðåõòî÷å÷íûõ óðàâíåíèé. Åå ìàòðèöàäèàãîíàëüíîé. Ýòà ñèñòåìà ðåøàåòñÿ ìåòîäîì ïðîãîíêè. Ìîæíî äîêàçàòü,îáðàçîì, ðåøåíèå äàííîé ñèñòåìû 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= i+1+ ψin ,2τhãäåψin =n+1un+1+ un+1un+1− unii+1 − 2uii−1i−+ fin+1h2τ(26)(27)Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòèÇàäà÷à. Ïîêàçàòü, ÷òî ψin èçÐåøåíèå.

Ðàçëîæèìun+1i±1èuni(27) åñòü65O(τ + h2 ).â ðÿä Òåéëîðà:n+1n+1un+1± ux,ih + un+1i±1 = uixx,ih3h2± un+1+ O(h4 )xxx,i262uni = un+1− un+1it,i τ + O(τ )Ïîäñòàâèì ýòè ðàçëîæåíèÿ â ôîðìóëó (27). Ïîëó÷èì:n+1n+1) + O(τ + h2 ) = O(τ + h2 )ψin = (−un+1t,i + uxx,i + fiÇàìåòèì, ÷òî:n+1z0n+1 = zN= zi0 = 0,Ïóñòü∃i0 ,i = 0, . . .

, N(28)òàêîé ÷òî:|zin+1| = max |zin+1 | = ||z n+1 ||C01≤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:||z 0 ||C = 0.Òàê êàê||ψ k || ≤ M (τ + h2 ),||zn+1||C ≤ MNXãäå M íå çàâèñèò îòτè h, òî:τ (τ + h2 )k=0Òàêèì 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 ñõåìà àñáîëþòíî ñõîäèòñÿn+1(èìååì àáñîëþòíóþ ñõîäèìîñòü ïåðâîãî ïîðÿäêà ïî τ è âòîðîãî ïîðÿäêà ïî h). Åñëè y0=n+1yN = 0, òî:NXn+1||y ||C ≤ ||u0 ||C +τ ||f k ||Ck=0Òàêèì îáðàçîì, ïîëó÷àåì óñòîé÷èâîñòü ÷èñòî íåÿâíîé ðàçíîñòíîé ñõåìû ïî íà÷àëüíîìó ïðèáëèæåíèþ è ïðàâîé ÷àñòè.Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòè66Ñèììåòðè÷íàÿ ðàçíîñòíàÿ ñõåìà (ñõåìà Êðàíêà-Íèêîëüñîíà)Îáîçíà÷èì ÷åðåçmyxx,iâòîðóþ ðàçíîñòíóþ ïðîèçâîäíóþ ïî ïðîñòðàíñòâåííîé ïåðåìåííîé:myxx,i=mmyi+1− 2yim − yi−1h2Ðàçíîñòíàÿ ñõåìà èìååò âèä:yin+1 − yinn+1n= 0.5(yxx,i) + f (xi , tn + 0.5τ )+ yxx,iτn+1y0n+1 = µ1 (tn+1 ), yN= µ2 (tn+1 ),yi0 = u0 (xi ),Ââåäåì ïîãðåøíîñòü:zin = yin − uni .n+1+ unxx,i ) −ψin = 0.5(uxx,iun+1i±1èuni(30)(31)Òîãäà äëÿ ïîãðåøíîñòè èìååì:n+1z0n+1 = zN= 0,Ðåøåíèå.

Ðàçëîæèìtn+1 ∈ ω txi ∈ ω hzin+1 − zinn+1n= 0.5(zxx,i+ zxx,i) + ψin ,τÇàäà÷à. Ïîêàçàòü, ÷òî ψin èç(29)(34) åñòüzi0 = 0,(xi , tn+1 ) ∈ ωτ hi = 0, . . . , Nn+ 21(34)O(τ 2 + h2 ).(xi , tn+ 1 ):21 n+ 1τ+ utt,i 2+ O(τ 3 )2 221 n+ 1 τ 2n+ 1 τ− ut,i 2 + utt,i 2+ O(τ 3 )2 22n+ 21uni = ui(33)un+1− unii+ f (xi , tn + 0.5τ )τâ ðÿä Òåéëîðà â îêðåñòíîñòè òî÷êèun+1= uii(32) τ 2n+ 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 âñå âõîæäåíèÿ ôóíêöèè âðÿä Òåéëîðà.

Ïðèâîäÿ ïîäîáíûå ñëàãàåìûå, ïîëó÷èì:nuxx,i= unxx,i + unxxxx,iÏðèìåíèì ýòî ðàçëîæåíèå êh2+ O(h4 )12n+1uxx,i, à çàòåì ïðîâåäåì åùå îäíî ðàçëîæåíèå â ðÿä Òåéëîðà â òî÷êå(xi , tn+ 1 ):2h2+ O(h4 ) =122h2 τn+ 1 τn+ 12 hn+ 12+ uxxt,i2 + uxxxx,i+ uxxxxt,i· + O(τ 2 + h4 )21212 2n+1n+1uxx,i= un+1xx,i + uxxxx,in+ 1= uxx,i2Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòèÒî æå ñàìîå ïðîäåëàåì è ñ67n:uxx,ih2+ O(h4 ) =12unxx,i = unxx,i + unxxxx,in+ 12τh2 τn+ 12 hn+ 12+ uxxxx,i− uxxxxt,i· + O(τ 2 + h4 )21212 2nâûðàæåíèå äëÿ ψi è ó÷òåì óðàâíåíèå òåïëîïðîâîäíîñòè:n+ 1= uxx,i2 − uxxt,i2Ïîäñòàâèì ýòè ðàçëîæåíèÿ ân+ 21ψin = (−ut,in+ 1n+ 12+ uxx,i2 + fin+ 12) + uxxxx,ih2+ O(τ 2 + h4 ) = O(τ 2 + h2 )12Çàäà÷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, àÇØË.

Ðåøåíèåì äàííîé çàäà÷è ÿâëÿþòñÿ ñîáñòâåííûå çíà÷åíèÿλkλ- ñîáñòâåííûå çíà÷åíèÿè ñîáñòâåííûå ôóíêöèèuk (x):λk = (πk)2 ,k = 1, 2, . . .0 < λ1 < λ2 < · · · < λn < . . .uk (x) = C sin(πkx)Ðàññìîòðèì ëèíåéíîå ïðîñòðàíñòâîR1L2 ôóíêöèé, èíòåãðèðóåìûõ ñ êâàäðàòîì íà [0, 1] ( f 2 (x)dx <0∞).Ââåäåì ñêàëÿðíîå ïðîèçâåäåíèå è íîðìó âL2 :Z1∀f, g ∈ L2 : (f, g) =f (x)g(x)dx0Z1||f ||L2 = √C = 2, 21f 2 (x)dx0òîãäà(uk , ul ) = δkl ,{uk (x)}∞k=1- îðòîíîðìèðîâàííûé áàçèñ â L2 .∞PÒàêèì îáðàçîì, ëþáóþ ôóíêöèþ f ∈ L2 ìîæíî ïðåäñòàâèòü â âèäå: f (x) =Ck uk (x), ãäå Ckk=1- êîýôôèöèåíòû Ôóðüå. Èìååò ìåñòî ðàâåíñòâî Ïàðñåâàëÿ:Âîçüìåìòî åñòü||f ||2L2=∞Xk=1Ck2Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòè68Ðàññìîòðèì äèñêðåòíûé àíàëîã çàäà÷è Øòóðìà-Ëèóâèëëÿ: yxx,i + λyi = 0, i = 1, .

. . , N − 1,y0 = yN = 0,yi íå ðàâíû òîæäåñòâåííî 0;(35)Ïîäñòàâèì â ïåðâîå óðàâíåíèå ïðåäñòàâëåíèå âòîðîé ðàçíîñòíîé ïðîèçâîäíîé:yi+1 − 2yi + yi−1 + λh2 yi = 0yi+1 + yi−1 = (2 − λh2 )yiÁóäåì èñêàòüyi = y(xi )â âèäåsin(αxi ), α ∈ R.yi+1 = sin(α(xi + h)),Òîãäà:yi−1 = sin(α(xi − h))yi+1 + yi−1 = sin(α(xi + h)) + sin(α(xi − h)) = 2 sin(αxi ) cos(αh)2 sin(αxi ) cos(αh) = (2 − λh2 ) sin(αxi )2 cos(αh) = (2 − λh2 )4 sin22(1 − cos(αh))=λ=h2h2Èç óñëîâèÿy0 = yN = 0αh2(36)èìååì:sin α = 0,k∈Zα = πk,Èòàê, ìû íàøëè ñîáñòâåííûå çíà÷åíèÿ è ñîáñòâåííûå ôóíêöèè äèñêðåòíîé çàäà÷è ØòóðìàËèóâèëëÿ:λk =4αhsin22h2√2 sin(πkxi ), i = 0, . .

. , N, k = 1, . . . , N − 1√Êîíñòàíòó âûáèðàåì ðàâíîé2 èç ñîîáðàæåíèé îðòîíîðìèðîâàííîñòè.Ââåäåì H - ëèíåéíîå ïðîñòðàíñòâî ñåòî÷íûõ ôóíêöèè: dim H = N − 1, ∀u ∈ H : u0 = uN = 0.yk (xi ) =Îïðåäåëèì ñêàëÿðíîå ïðîèçâåäåíèå è íîðìó â H:∀u, v ∈ H : (u, v) =N−1Xui vi hi=1||u||H =NX! 12u2i hi=1N −1Ïóñòü (yk , yl ) = δk l , òîãäà {yk (xi )}k=1 - îðòîíîðìèðîâàííûé áàçèñ â H, è ëþáóþ ôóíêöèþìîæíî ïðåäñòàâèòü â âèäå:u(xi ) =N−1Xk=1Ck yk (xi ),i = 0, . .

. , Nu∈HÐàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòè69Èìååò ìåñòî òàêæå è ðàâåíñòâî Ïàðñåâàëÿ:||u||2H=N−1XCk2k=1Ïóñòüµk (xi ) ≡ yk (xi ),k = 1, . . . , N − 1,ÇØË (35). Âåðíåìñÿ ê (32) - (34). Ðàçëîæèìzin=i = 0, . . . , N, - ñîáñòâåííûåzin è ψin ïî áàçèñó èç µn :N−1Xôóíêöèè äèñêðåòíîék (tn )µk (xn )1ψin=N−1Xψ (k) (tn )µk (xn )1Ïîäñòàâèì ýòè ðàçëîæåíèÿ â (32):N−1Xµk (xi )(ck (tn+1 ) − ck (tn )) =k=1= 0.5τN−1XN−1Xk=11(µk )xx,i (ck (tn+1 ) + ck (tn )) + τψ (k) (tn )µk (xn )ck (tn+1 ) − ck (tn )+ 0, 5λk (ck (tn+1 ) + ck (tn )) = ψ (k) (tn )τn = 0, 1, .

. . , k = 1, N − 1, ck (0) = (r(0), µk ) = 0Ðàçðåøèì óðàâíåíèå îòíîñèòåëüíî (n+1)-ãî ñëîÿ.ck (tn+1 ) =Ïîëîæèìqk =1 − 0, 5τ λkτck (tn ) +ψ (k) (tn )1 + 0, 5τ λk1 + 0, 5τ λk1−0,5τ λk1+0,5τ λkck (tn+1 ) = qk ck (tk ) +τψ (k) (tn )1 + 0, 5τ λkÒîãäà:zin+1=N−1Xck (tn+1 )µ(k) (xi ) =k=1N−1Xk=1kqk ck (tn )µ (xi ) +N−1Xk=1τψ k (tn )µk (xi ) = vin+1 + win+11 + 0.5τ λkÎ÷åâèäíî,kz n+1 k ≤ kv n+1 k + kwn+1 kÎöåíèìkv n+1 k,èñïîëüçóÿ ðàâåíñòâî Ïàðñåâàëÿ.(37)Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòè|qk | < 1 ⇒ kvn+1 2k =N−1Xqk2 c2k (tn )≤N−1Xc2k (tn ) = kz n k270(38)k=1k=1Àíàëîãè÷íî,kwn+1 k2 ≤ τ 2 kψ n k2(39)Ó÷èòûâàÿ (38) è (39) íåðàâåíñòâî (37) ïðèìåò âèä:kzn+1nn0k ≤ kz k + τ kψ k ≤ kz k +N−1Xτ kψ k kk=1Èç ðàíåå ðåøåííîé çàäà÷è:kψ k k ≤ M (τ 2 + h2 ) ⇒kz n+1 k ≤ M T (τ 2 + h2 ) → 0ãäåMèTíå çàâèñÿò îòτèïðèτ, h → 0,0 < M = const(40)h.Ðàçíîñòíàÿ ñõåìà ñ âåñàìè.