Численные методы. Ионкин (2012) (v1.1) (косяки есть) (1160435), страница 11
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Ðàçíîñòíûå ñõåìû äëÿ óðàâíåíèÿ Ïóàññîíà.(Çàäà÷à Äèðèõëå)Ðàññìàòðèâàåì óðàâíåíèå:∂ 2U ∂ 2U+= f (x1 , x2 );∂x21∂x22U (x1 , x2 )íàD = {(x1 , x2 );ãðàíèöå Ã(x1 , x2 ) ∈ (D)(4.45)= µ(x1 , x2 )(4.46)0 < x1 < l1 , 0 < x2 < l2 }Íà÷èíàåì ðåøåíèå ñ ïîñòðîåíèÿ ñåòêè:(i)(j)ωh = { (x1 , xi ) = xij ,(i)x1 = ih;N1 h1 = l1 ;93(j)x2 = jh2 ;N2 h2 = l2 }Íàïîìíèì:∂ 2u ∂ 2u+= f (x1 , x2 ) (x1 , x2 ) ∈ D∂x21 ∂x22(4.47)u(x1 , x2 )|r = µ(x1 , x2 )(4.48)Ðèñ. 4.7:2 −11 −11 −1Γn = {x0,j }1N2 −1 ∪ {xN1 ,j }N∪ {xi,0 }N∪ {xi,N2 }N111ω̄n = ωn êðóæî÷êè ∪ Γn êâàäðàòèêè âñå óçëûÂòîðàÿ ïðîèçâîäíàÿ:yx̄1 x1 ;i,j =yi+1,j − 2yi,j + yi−1,jh21(i)(i)fi,j = f (x̄1 , x2 ) = f (xi,j )yx̄1 x1 ;i,j + yx̄2 x2 ;i,j = fi,j xi,j ∈ ωh(4.49) ðàçíîñòíàÿ àïïðîêñèìàöèÿÃðàíè÷íîå óñëîâèå ïåðâîãî ðîäà ïîýòîìó àïïðîêñèìèðóåì òî÷íî 1 ≤ i ≤ N1 − 11 ≤ j ≤ N2 − 194yi,j |Γn = µi,j(4.50)Ðèñ.
4.8: øàáëîí ðàçíîñòíîé ñõåìû òèïà ¾êðåñò¿Èçó÷èì 5 âîïðîñîâ: ñóùåñòâîâàíèå-åäèíñòâåííîñòü ðåøåíèÿ, ïîãðåøíîñòü àïïðîêñèìàöèè (ñàìûé ë¼ãêèé). . .Ïåðâîå ñóæäåíèå î ïîãðåøíîñòè:zi,j = yi,j − ui,jzx̄1 x1 ;i,j + zx̄2 x2 ;i,j = −Ψi,j(4.51)zi,j |Γn = 0(4.52)Ψi,j = ux̄1 x1 ;i,j + ux̄2 x2 ;i,j − fi,j(4.53) íåâÿçêà.Çàäà÷à:Ψi,j = O(h21 + h22 )á22Ψi,j = ∂∂u x21 + h12 (...)Ïîêàçàòü ÷òîÏîäñêàçêà:(u(x1 , x2 )95∈ C 4 (D̄)) 6. Ðàçðåøèìîñòü ðàçíîñòíîé çàäà÷è. Ñõîäèìîñòüðàçíîñòíîé ñõåìûÐàñïèøåì 4.49 îòíîñèòåëüíî öåíòðàëüíîãî óçëà, çàïèñàâ êîîðäèíàòíî:222yi−1,j + yi+1,j yi,j+1 + yi,j−1+− fi,j 1 ≤ i ≤ N1 − 1, 1 ≤ j ≤ N2 − 1( 2+)yi,j =h1 h2h21h22(4.54)yi,j |Γn = µi,jßâíî ìû íè÷åãî âû÷èñëèòü íå ìîæåì. Äëÿ äîêàçàòåëüñòâà ñóùåñòâîâàíèÿáóäåì îïèðàòüñÿ íà àïïàðàò àëãåáðû.
(Ðàíãè ðàñøèðåííûõ ìàòðèö äîëæíûñîâïàäàòü ñ ÷èñëîì íåèçâåñòíûõ; äðóãèìè ñëîâàìè, îäíîðîäíàÿ ñèñòåìà èìååòòîëüêî òðèâèàëüíîå ðåøåíèå çíà÷èò, îïðåäåëèòåëü áóäåò îòëè÷åí îò íóëÿ.)Çàïèñûâàåì ñîîòâåòñòâóþùóþ îäíîðîäíóþ çàäà÷ó.vi−1,j + vi+1,j vi,j+1 + vi,j−12 22)vi,j =+1 ≤ i ≤ N1 −1, 1 ≤ j ≤ N2 −1( 2+h1 h2h21h22(4.55)vi,j |Γn = 0Ìàòðèöà ïîëó÷èëàñü áû ÷ðåçâû÷àéíî ëþáîïûòíîãî âèäà: ìàòðèöà èç òð¼õäèàãîíàëüíûõ áëîêîâ. Àëãîðèòì ðåøåíèÿ ïîòîì. À ñåé÷àñ âñ¼-òàêè ñóùåñòâîâàíèåè åäèíñòâåííîñòü.Òåîðåìà 1:Ñèñòåìà 4.55 èìååò òîëüêî òðèâèàëüíîå ðåøåíèå:vi,j === 0 xi,j ∈ ω̄nÄîêàçàòåëüñòâî:Äîêàçàòåëüñòâî íà îñíîâå ïðèíöèïà ìàêñèìóìà.∃Ýòîóçåëxi0 ,j0 : |vi0 ,j0 | =maxîò 0 äî Nn i,j|vi,j | = kvkC ; vi0 ,j0 6= 0.C(D̄)!Ñðåäè ýòèõ óçëîâ âûáåðåì îäèí ñ äâóìÿ ñâîéñòâàìè:1.|vi0 ,j0 | = kvkC ;96À ýòî íîðìà - íå òà!2.
Õîòÿ áû â îäíîì èç ÷åòûð¼õ ñîñåäíèõ óçëîâ:|vi,j | < |vi0 ,j0 |.Îí ñóùåñòâóåò, ïî êðàéíåé ìåðå ïîòîìó, ÷òî íà ãðàíèöàõ íîëü, à â óçëå íåò.Íàøå óðàâíåíèå â óçëå:222vi ,j +1 + vi0 ,j0 −1vi −1,j + vi +1,j( 2+)vi0 ,j0 = 0 0 2 0 0 + 0 0h1 h2h1h22Ìàæîðèðóåì:(222|vi0 −1,j0 | + |vi0 +1,j0 | |vi0 ,j0 +1 | + |vi0 ,j0 −1 |++)|v|≤i,j00h21 h2h21h22Èç ÷åòûð¼õ ñëàãàåìûõ åñòü õîòÿ áû îäíî, â êîòîðîì çíà÷åíèå ñòðîãî ìåíüøå,âåäü ìû ñòàâèëè ñâîéñòâî óçëà 2.222222)kvkC < ( 2 +)kvkC( 2+h1 h2h1 h2 ïðîòèâîðå÷èå.
Äîêàçàëè, ÷òî îäíîðîäíàÿ ñèñòåìà èìååò òîëüêî òðèâèàëüíîåðåøåíèå.÷òä.Ñëåäñòâèå:Âîçâðàùàåìñÿ ê èñõîäíîé ñõåìå 4.49 - 4.50. Ýòà ñõåìà èìååò åäèíñòâåííîåðåøåíèå ïðè ëþáûõµè 0f .Êëþ÷åâîé âîïðîñ ñõîäèìîñòü è óñòîé÷èâîñòü.Ñðàçó áóäåì äîêàçûâàòü ñõîäèìîñòü, ïîòîìó ÷òî îíà èòàê åñòü ïî òåîðåìåÊýëè. Äîêàçûâàòü áóäåì â íîðìåC.Äàæå àñïèðàíòû ïðè ñäà÷å êàíäìèíèìóìà îò ýòîãî âîþò.Ïîãðåøíîñòü îöåíèâàåòñÿ íåñëîæíî, èìååò âòîðîé ïîðÿäîê. Èç âñåãî, ÷òîkzkC ≤ M (h21 + h22 ),h1 → 0, h2 → 0: kzkC = ky − ukC → 0áûëî, çíàåì, ÷òî åñëè ïîëó÷èì îöåíêó:h1 , h2 ,òî ïðèãäåÌîæåò ñðàçó ðàçðåøèì îòíîñèòåëüíî öåíòðàëüíîãî óçëà?yi−1,j − 2yi,j + yi+1,j yi,j−1 − 2yi,j + yi,j+1+= fi,jh21h2297Míå çàâèñèò îòÄëÿ áîëüøåé îáùíîñòè ââåä¼ì îïåðàòîð:Lh vi,j = (2vi−1,j + vi+1,j vi,j−1 + vi,j+12+)v−−i,jh21 h22h21h21xi,j ∈ ωhËåììà:vi,j ≥ 0 íà ãðàíèöå xi,j ∈ Γh ,≥ 0 âåçäå xi,j ∈ ω̄h .
(ïðèíöèï max)Ïóñòüvi,jè ïóñòüLh vi,j ≥ 0âíóòðèxi,j ∈ ωh .ÒîãäàÄîêàçàòåëüñòâî:Êàê è ïîëîæåíî, äîêàçûâàåì îò ïðîòèâíîãî.∃óçåëxi0 ,j0 : vi0 ,j0 < 0.Âûáåðåì, îïÿòü-òàêè, óçåë ñ äâóìÿ óñëîâèÿìè:1.2.vi0 ,j0 < 0;vi0 ,j0 < vñîñåä .Ïåðåïèøåì â ýòîì óçëå.Lh vi0 ,j0 =vi0 ,j0 − vi0 −1,j0 vi0 ,j0 − vi0 +1,j0 vi0 ,j0 − vi0 ,j0 −1 vi0 ,j0 − vi0 ,j0 +1+++h21h21h22h22Îäíî èç ñëàãàåìûõ ñòðîãî ìåíüøå íóëÿ ïî âòîðîìó óñëîâèþ. ÒîãäàLh vi0 ,j0 < 0 ïðîòèâîðå÷èå.Ïîäîéä¼ì ê òåîðåìå ñðàâíåíèÿ, òàì áóäåò ìàæîðàíòà.Ñëåäñòâèå 1 (äèñêðåòíûé àíàëîã òåîðåìû ñðàâíåíèÿ):(îáà Ln yi,j = φi,j(4.56)Ln Yi,j = Φi,j(4.57)xi,j ∈ ωn ).
yi,j |Γn , Yi,j |ΓnÓòâåðæäàåòñÿ: åñëè|yi,j | ≤ Yi,jçàäàíî.íà ãðàíèöåxi,j ∈ Γn . (Y ìàæîðàíòà, îíàâñåãäà íåîòðèöàòåëüíà, ìîæíî òàê ñðàâíèâàòü ñ ìîäóëåì), è åñëèω,òî|yi,j | ≤ Yi,j âåçäå.98|φi,j | ≤ Φi,jíà×òî ýòî íàì äàåò: ìàæîðàíòó âûáåðåì òàê, ÷òî îöåíêó ñäåëàåì ÷åðåç øàãè, èïîëó÷èì àïðèîðíóþ îöåíêó, êîòîðàÿ äîêàçûâàåò ñõîäèìîñòü è ÿâëÿåòñÿ îöåíêîéóñòîé÷èâîñòè.vi,j = Yi,j − yi,j ; wi,j = Yi,j + yi,j .ÏîäåéñòâóåìLh :Lh vi,j = Φi,j − φi,j ≥ 0vi,j ≥ 0, xi,j ∈ ωh .(ïîòîìó ÷òîvi,j ≥ 0, xi,j ∈ Γn ) ñëåäîâàòåëüíîLh wi,j = Φi,j + φi,j ≥ 0wi,j ≥ 0, xi,j ∈ ωh .(ïîòîìó ÷òîwi,j ≥ 0, xi,j ∈ Γn ) ñëåäîâàòåëüíîÝòî îçíà÷àåò, ÷òî|yi,j | ≤ Yi,j .Çàéì¼ìñÿ ïîñòðîåíèåì ìàæîðàíò.Ïåðåïèøåì çàäà÷ó äëÿz÷åðåç îïåðàòîð.Lh zi,j = Ψi,j .Çíàê çäåñü äåéñòâèòåëüíî ïîìåíÿåòñÿ íà ïðîòèâîïîëîæíûé, ýòî íå îïå÷àòêà.Íàøà çàäà÷à: Lh zi,j = Ψi,j xi,j ∈ ωn(4.58)zi,j |Γn = 0(i)(j)Yi,j = K(l12 + l22 − (x1 )2 − (x2 )2 )K > 0 êîíñòàíòàYi,j ≥ 0 xi,j ∈ ω̄h .(âûáåðåì ïîòîì).Çàäà÷à:Ïîêàçàòü, ÷òîLh Yi,j = 4KÐåøåíèå:Lh Yij = K1 , K1 > 0.22(i) 2(j) 2Ïðåäñòàâèì Yij = (l1 + l2 − (x1 ) − (x2 ) )K, K > 0ÏîëîæèìYij Lh ≥ 0, xij ∈ ω̄ ⇒99(4.59)Ìû áåðåì îò êàæäîãî ïî äâîéêå, à äâå êîíñòàíòû îáíóëÿåì.Yij Lh = 4KÏîëîæèìk|ψ||c = 4KLn Yi,j = kΨkC , xi,j ∈ ωh ; 4K = kΨkCYi,j |Γn ≥ 0, xi,j ∈ ΓnÏîïàäàåì â óñëîâèÿ ñëåäñòâèÿ 1, ïîýòîìó0 ≤ Yi,j ≤M=íàxi,j ∈ ω̄h .l12 +l224 kΨkCl12 +l224 - Ïîëó÷àåòñÿ, ÷òîMçàâèñèò òîëüêî îò ãðàíèö îáëàñòè.kzkC ≤Ýòî îçíà÷àåò óñòîé÷èâîñòü äëÿ4.49|zi,j | ≤ Yi,j(4.60)+yi,j |Γn = 0y:l12 + l22kΨkC4âìåñòîΨ(4.61)ñòàíåòφ: â òî÷íîñòè òà çàäà÷à, êîòîðóþ ìû èññëåäîâàëè äëÿz.Çíà÷èò,l12 + l22kykC ≤kφkC4(4.62) îíà è îçíà÷àåò óñòîé÷èâîñòü.Òåîðåìà 2:u(x1 , x2 ) ∈ C 4 (D̄)Òîãäà ðàçíîñòíàÿ ñõåìà 4.49 - 4.50 ñõîäèòñÿ ê ðåøåíèþ çàäà÷è 4.47 - 4.48 ñîâòîðûì ïîðÿäêîì ïîh1èh2(èëè èìååò âòîðîé ïîðÿäîê òî÷íîñòè).Äîêàçàòåëüñòâî:kΨkC ≤ M (h21 + h22 ), M > 0 íå çàâèñèò îò h1 , h2 .l2 ,l222Ïîëó÷àåì: kzkC ≤ M1 (h1 + h2 ).
M1 = M 1 2 íå4À òàê êàê kzkC = ky − ukC âîò è äîêàçàëè.÷òä.100çàâèñèò îòh1 , h2 .Åäèíñòâåííûé çàâèñøèé âîïðîñ êàê ðåøàòü-òî? Ïîêàæåì íà ñëåäóþùåéëåêöèè: åñòü è ïðÿìûå ìåòîäû, íî ñàìûå ðàñïðîñòðàí¼ííûå èòåðàöèîííûå.(Ìåòîä Ñàìàðñêîãî ïðî ïåðåìåííûé òðåóãîëüíèê)Ìåòîäû ðåøåíèÿ ðàçíîñòíîé çàäà÷è Äèðèõëå.Ðàçâåðíóòûé âèä:yi−1 − 2yij + yi+1,j yi,j−1 − 2yi,j + yi,j+1+= fi,j , i = 1, . . .
, N1 −1, j = 1, . . . , N2 −1h21h22(4.63)yij |Γn = µij , xij ∈ Γnò.ê íåîáõîäèìà õîðîøàÿ òî÷íîñòü(4.64)(h1 → 0, h2 → 0),òî ÷åì ìåíüøå ìû âîçü-ìåì øàã, òåì âûøå áóäåò òî÷íîñòü.Âîçìîæíà ñèòóàöèÿ, êîãäà íåîáõîäèìî ðåøåíèå îïåðàòîðà Ëàïëàññà ñ 1000 óðàâíåíèé. Âñòàåò âîïðîñ: "Êàê ðåøàòü"? Âåçäå ïðèìåíÿòü - ýòî íåðàöèîíàëüíî. Ïðèìåíåíèå ìåòîäà Ãàóññà - íåðàöèîíàëüíî. (Ñëîæíîñòü∼ n3 ).Ñàìîå øèðîêîå ïðèìåíåíèå ïîëó÷èë èòåðàöèîíûé ìåòîä.Óðàâíåíèå äëÿ öåíòðàëüíîãî óçëà:((S)yij − S -ÿ22yi−1,j + yi+1,j yi,j−1 + yi,j+1+ 2 )yij =+− fij ,2h1 h2h21h22èòåðàöèÿ.Ìåòîä ïîïåðåìåííîé èòåðàöèè.(s)(s)(s)(s)2 (s+1) yi−1 + yi+1,j yi,j−1 + yi,j+12( 2 + 2 )yij=+− fij ,h1 h2h21h22yij0 -S = 0, 1, . . .h = h1 = h2 ⇒ h0 (ε) ∼ O(h−2 ) ⇒(4.65)íà÷àëüíîå ïðèáëèæåíèåÏðèìåìñõîäèìîñòü ÷ðåçâû÷àéíî ìåäëåííàÿ.Ýòî íå ýêîíîìè÷íûé ìåòîä.Ìåòîä Çåéäåëÿ (íåÿâíûé)(s+1)(s)(s+1)(s)22 (s+1) yi−1,j + yi+1,j yi,j−1 + yi,j+1+)y+− fij ,=h21 h22 ijh21h22(0)yi,j -çàäàí, s = 0, 1, 2, .
. .101(4.66)Ðèñ. 4.9:Ìû äâèæåìñÿ èç 1 â 2.y1,j : j = 1, . . . , N2 − 1y2,j : j = 1, . . . , N2 − 1∗ÑËÀÓ: Ay = ϕ - ðàçíîñòíàÿ çàäà÷à. A = A > 0Ïðåäñòàâèì A = R1 + R2 , ãäå R1 -ìàòðèöà, èìåþùàÿ íèæíþþ òðåóãîëüíóþ ôîðìóñ 0.5ai íà äèàãîíàëè. K2 -ìàòðèöà, èìåþùàÿ âåðõíþþ òðåóãîëüíóþ ôîðìó.y (s+1) − y (s)+ Ay (s) = ϕ(E + wR1 )(E + wR2 )ττãäå τ ≥ 0, w > 0, w >4 íåîáõîäèìîå óñëîâèå ñõîäèìîñòè.Íà÷àëüíîå óñëîâèå y0 -çàäàíî, s = 0, 1, . .
.Îáîçíà÷èì:(4.67)y (s+1) − y (s)1)(e + wR2 )= W (s+1)τ(s+1)2)(E + wR1 )W= ϕ − Ay (s)3)V (s+1) =y (s+1) − y (s)τ→ W (s+1)(s+1)Èç 2-ãî → V(s+1)Èç 3-ãî → y= y (s) + τ v (s+1) ⇒ n0 (ε) = O(h−1 ). Ýòîò ìåòîä ÿâëÿåòñÿ âåäóùèì.Èç 1-ãî102 7. Îñíîâíûå ïîíÿòèÿ òåîðèè ðàçíîñòíûõ ñõåì: àïïðîêñèìàöèÿ, óñòîé÷èâîñòü, ñõîäèìîñòüÑëåäóþùèå âûñêàçûâàíèÿ ïðèñóùå âñåì ëèíåéíûì çàäà÷àì ìàòåìàòè÷åñêîéôèçèêè.LU (x) = f (x), x ∈ G(4.68)ãäå L-ëèíåéíûé äèôôåðåíöèàëüíûé îïåðàòîð.x = (x1 , .
. . , xm )-ëèíåéíûéâåêòîð.Íà÷èíàåì ñ ïîñòðîåíèÿ ñåòêè.G → Gh ,h-íåêîòîðàÿ íîðìà øàãîâ (îáîáùåííàÿ õàðàêòåðèñòèêà). Íà ïðàêòè-êå âûáîð ñåòêè - ñåðüåçíûé âîïðîñ.Ââîäèì ñåòî÷íûå ôóíêöèè.yh , LH .Ðàçíîñòíàÿ ñõåìà:LH yh (x) = ϕn (x), x ∈ Gn , h =| h |(4.69)Àïïðîêñèìèðóåì èñõîäíóþ äèôôåðåíöèàëüíóþ çàäà÷ó. Íàó÷èìñÿ èçìåðÿòüðàññòîÿíèå ìåæäó ôóíêöèÿìè èç ðàçíîñòíûõ íîðìèðîâàííûõ ïðîñòðàíñòâ.y(x) ∈ B0 , x ∈ Gyh (x) ∈ Bh , x ∈ GhPh : Bi → Bh - îïåðàòîð ïðîåêòèðîâàíèÿ.Ph U = Uh(x) - íà ñåòêå x ∈ GhÂâîäèì íîðìû, ÷òîáû èññëåäîâàòü ñõîäèìîñòü íà íîðìå.k.k0-âB0 ; k.kh- âBhÍîðìû äîëæíû áûòü ñîãëàñîâàíûG : 0 ≤ x ≤ 1.0, 1, .
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