Численные методы. Ионкин (2009), страница 10

Ïîãðåøíîñòü àïïðîêñèìàöèè.Ïîñòðîèì äëÿ çàäà÷è (1) ðàçíîñòíóþ ñõåìó:yin+1 − yinn+1n+ (1 − σ)yxx,i= σyxx,i+ φni ∈ ωτ hτy0n+1 = µ1 (tn+1 ), tn+1 ∈ ωτn+1yN= µ2 (tn+1 ), tn+1 ∈ ωτyi0 = u0 (xi ), xi ∈ ωhσ ∈ R,Äëÿ ðàçëè÷íûõσ0≤σ≤1ïîëó÷àåì:1.σ=0- ÿâíàÿ ðàçíîñòíàÿ ñõåìà.2.σ=1- ÷èñòî íåÿâíàÿ ðàçíîñòíàÿ ñõåìà.3.σ = 0.54.σ 6= 0, 1, 0.5- ñèììåòðè÷íàÿ ðàçíîñòíàÿ ñõåìà.- íåÿâíàÿ ðàçíîñòíàÿ ñõåìà.Ââåäåì ïîãðåøíîñòüzin = yin − uni .zin+1 − zinn+1n= σzxx,i+ (1 − σ)zxx,i+ ψinτn+1= zi0 = 0z0n+1 = zNÏîãðåøíîñòü àïïðîêñèìàöèè ðàçíîñòíîé ñõåìû (41) íà ðåøåíèè:(41)Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòènψin = σun+1xx,i + (1 − σ)uxx,i −71un+1− unii+ φniτ(42)u0 =∂u, u̇ = ∂u.

Ïóñòü ôóíêöèÿ u(x, t) øåñòü ðàç íåïðåðûâíî äèôôåðåíöèðóåìà∂x∂tïî x è òðè ðàçà ïî t. Ðàçëîæèì åå ïî ôîðìóëå Òåéëîðà â îêðåñòíîñòè òî÷êè (xi , tn+ 1 ):Îáîçíà÷èì22h 00u +2 ih2= ui − hu0i + u00i −2ui+1 = ui + hu0i +ui−13h 000u +6 ih3 000u +6 i4h 0000u + ...24 ih4 0000u + ...24 iττ2τ31) +1) +1) +u̇(tü(tüi (tn+ 1 ) + . . .un+1=u(tiiin+ 2n+ 2n+ 2i22848ττ2τ3uni = ui (tn+ 1 ) − u̇i (tn+ 1 ) + üi (tn+ 1 ) − üi (tn+ 1 ) + . . .22222848h2 0000ui+1 − 2ui + ui−100=u+ui + O(h4 )i2h12uxx,i =un+1i= u̇i (tn+ 1 + O(τ 2 )2τÂîñïîëüçóåìñÿ íåðàâåíñòâîìτ h2 ≤τ 2 +h4:2τ 00 h2 0000=+ u̇i + ui + O(h4 ) + O(τ 2 ))+212τh2(1 − σ)(u00i − u̇00i + u0000+ O(h4 ) + O(τ 2 ))−212 i˙ i + φn + O(τ 2 + h4 ) =(u)ψi00σ(u00ii(u00i − u̇i + φni ) + (σ − 0.5)τ u̇00 +Ïðîäèôôèðåíöèðîâàâ óðàâíåíèåu00 − u̇ + f = 0h2 0000ui + O(τ 2 + h4 )12äâàæäû ïî x, ïîëó÷èì:u0000 − u̇00 + f 00 = 0 ⇒ u0000 = u̇00 + f 00Ïîäñòàâèìu0000iâ ôîðìóëó ïîãðåøíîñòè àïïðîêñèìàöèè:ψin=u00i|− u̇i + f (xi , t{zn+ 21=0h2 00) − f (xi , tn+ 1 ) + φi − f (xi , tn+ 1 )+22} 12(σ − 0.5)τ +h2 00u̇ + O(t2 + h4 )12Òàêèì îáðàçîì, ïîðÿäîê ïîãðåøíîñòè àïïðîêñèìàöèè çàâèñèò îò ïàðàìåòðàìàöèè ôóíêöèè f:σè àïïðîêñè-Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòè1.σ = σ∗ =12−h212τ⇒h2 00f (xi , tn+ 1 )2212n24ψi = O(τ + h )φni = f (xi , tn+ 1 ) +2.σ = 0.5 ⇒φni = f (xi , tn+ 1 ) + O(h2 ) + O(τ 2 )2ψin3.= O(τ 2 + h2 )σ 6= σ ∗ , σ 6= 0.5 ⇒ φni = f (xi , tn ) + O(τ + h2 ) ⇒ ψin = O(τ + h2 )Ðàçíîñòíûå ìåòîäû äëÿ óðàâíåíèÿ ÏóàññîíàÐàññìîòðèì óðàâíåíèå Ïóàññîíà â îáëàñòè D:∂ 2u∂ 2u+= f (x1 , x2 )∂ 2 x1 ∂ 2 x2(x1 , x2 ) ∈ D, D = {(x1 , x2 ) : 0 < x1 < l1 ; 0 < x2 < l2 }x26Ãl2ÃÃD-l1Ãx1Ââåäåì íà îáëàñòè D ñåòêó:n(i)(j)(i)ωh = (x1 , x2 ), x1 = ih, i = 1, N1 − 1, h1 N1 = l1(j)x2 = jh2 , j = 1, N2 − 1, h2 N2 = l2oÇàìåòèì, ÷òî ïðè ýòîì ãðàíè÷íûå óçëû ìû íå ðàññìàòðèâàåì.x26Ãl2Ãh2c cc cDh1ÃÃ-l1x1Ãh - ãðàíè÷íûå óçëû.ÃhN2 −1N1 −1N1 −12 −1= {x0,j }Nj=1 ∪ {xN1 ,j }j=1 ∪ {xi,0 }i=1 ∪ {xi,N2 }i=1ωh = ωh ∪ Ãh72Ðàçíîñòíûå ñõåìû äëÿ óðàâíåíèÿ Ïóàññîíà (çàäà÷à Äèðèõëå)Ÿ273Ðàçíîñòíûå ñõåìû äëÿ óðàâíåíèÿ Ïóàññîíà (çàäà÷à Äèðèõëå)Çäåñü áóäåò ðèñóíîê ñ ôîðìóëèðîâêè çàäà÷è.G = {(x1 , x2 ) : 0 < x1 < l1 , 0 < x2 < l2 },G=G∪Γ22du du+= f (x1 , x2 ), (x1 , x2 ) ∈ Gdx21 dx22(1)U |r = µ(x1 , x2 )(2)Ââåäåì ñåòêó:(i)(j)(i)ωh = {(x1 , x2 ) : x1 = ih1 , i = 1, N1 − 1, N1 h1 = l1 ;(j)x2 = jh2 , j = 1, N2 − 1, N2 h2 = l2 },N1 −1N1 −1N2 −12 −1Γh = {x0,j }Nj=1 ∪ {xN1 ,j }j=1 ∪ {xi,0 }i=1 ∪ {xi,N2 }i=1ωh = ωh ∪ ΓhÏóñòüyij = y(xi1 , xj2 ), fij = f (xi1 , xj2 ).Çàïèøåì ðàçíîñòíóþ ñõåìó äëÿ çàäà÷è (1), (2):yx1 x1 ,ij + yx2 x2 ,ij = fij , (xi1 , xj2 ) ∈ ωh(3)yij |Γh = µ(xi1 , xj2 ), (xi1 , xj2 ) ∈ Γh(4)(3)è (4) ïðåäñòàâëÿþò ñîáîé ÑËÀÓ.

Ðàñïèøåì (3):yi+1,j − 2yij + yi−1,j yi,j+2 − 2yij + yi,j−1+= fiih21h22yij |Γh = µij ,Ïîãðåøíîñòü ðàçíîñòíîãî ðåøåíèÿi = 1, N1 − 1, j = 1, N2 − 1Zij = yij − Uijóäîâëåòâîðÿåò çàäà÷åZx1 x1 ,ij + Zx2 x2 ,ij = −ψij(5)Zij |Γh = 0Ïîêàæåì ñóùåñòâîâåíèå è åäèíñòâåííîñòü ðåøåíèÿ ñèñòåìû (3). Äîêàæåì, ÷òî ðåøåíèå, ñîîòâåòñòâóþùåå îäíîðîäíîé ñèñòåìå, òðèâèàëüíî.

Ñîîòâåòñòâåííî, ðåøåíèå íåîäíîðîäíîé ñèñòåìûñóùåñòâóåò è åäèíñòâåííî. Ïåðåïèøåì ñèñòåìó (3) â âèäå:(22Vi+1,j + Vi−1,j Vi,j+1 + Vi,j−1+ 2 ) ∗ Vi,j =+,2h1 h2h21h220 < i < N1 ,0 < j < N2 .Òåîðåìà. Ñèñòåìà(6) èìååò òîëüêî òðèâèàëüíîå ðåøåíèå.(6)Ñõîäèìîñòü ðàçíîñòíîé çàäà÷è Äèðèõëå74Äîêàçàòåëüñòâî.

Ïðåäïîëîæèì, ÷òî íàéäåòñÿ òàêîé óçåëxij ,ãäåvij 6= 0.Òîãäà∃i0 , j0 ,òàêèå÷òî:à)|Vi0 ,j0 | = max |Vij |0≤i≤N10≤j≤N2b) õîòÿ áû â îäíîì óçëå(i0 , j0 ± 1), (i0 ± 1, j0 )Ðàññìîòðèì ðàçíîñòíóþ ñõåìó â óçëå(áóäåò âûïîëíåíî|Vij | < |Vi0 ,j0 |i0 , j0 :22Vi +1,j + Vi −1,jVi ,j +1 + Vi0 ,j0 −1+ 2 ) ∗ Vi0 ,j0 = 0 0 2 0 0 + 0 02h1 h2h1h22Îöåíèì ïî ìîäóëþ çíà÷åíèå ëåâîé ÷àñòè óðàâíåíèÿ:(Òàê êàê222||Vi0 +1,j0 ||C 2||Vi0 ,j0 +1 ||C+ 2 ) ∗ |Vi0 ,j0 | ≤+2h1 h2h21h22|Vi0 ,j0 | = ||V ||C :(2222+ 2 ) ∗ ||V ||C < ( 2 + 2 ) ∗ ||V ||C2h1 h2h1 h2Ïðèøëè ê ïðîòèâîðå÷èþ.

Ñëåäîâàòåëüíî, ïðåäïîëîæåíèå íåâåðíî è òåîðåìà äîêàçàíà.Cëåäñòâèå. Ðàçíîñòíàÿ çàäà÷à èìååò åäèíñòâåííîå ðåøåíèå äëÿ ëþáûõ ôóíêöèé f è µ.Ÿ3Ñõîäèìîñòü ðàçíîñòíîé çàäà÷è ÄèðèõëåÐàññìîòðèì çàäà÷ó:Zx1 ,x1 ,ij + Zx2 ,x2 ,ij = −ψ,Zij |Γh = 0,xij ∈ ωh(1)xij ∈ ΓhÂâåäåì ðàçíîñòíûé îïåðàòîð:Lh Vij = (22Vi +1,j + Vi −1,jVi ,j +1 + Vi0 ,j0 −1+ 2 ) ∗ Vij − 0 0 2 0 0 + 0 0,2h1 h2h1h22xij ∈ ωhÓòâåðæäåíèå. Ïóñòü Vij ≥ 0, Xij ∈ Γh , Ln Vij ≥ 0, xij ∈ ωh .

Òîãäà Vij ≥ 0.Äîêàçàòåëüñòâî. Äîêàæåì îò ïðîòèâíîãî. Ïðåäïîëîæèì, ÷òî∃(i0 , j0 )òàêèå, ÷òî:a)|Vi0 ,j0 | = min |Vij |0≤i≤N10≤j≤N2b) õîòÿ áû â îäíîì óçëå(i0 , j0 ± 1), (i0 ± 1, j0 )áóäåò âûïîëíåíîVi0 ,j0 < VijÒîãäà:Lh Vi0 j0 =Vi0 ,j0 + Vi0 +1,j0 Vi0 ,j0 + Vi0 −1,j0 Vi0 ,j0 + Vi0 ,j0 +1 Vi0 ,j0 + Vi0 ,j0 −1+++h21h21h22h22Ñîãëàñíî óñëîâèþ, õîòÿ áû îäíî èç ýòèõ ñëàãàìûõ ìåíüøå 0. Ñëåäîâàòåëüíî ñóììà òîæå ìåíüøåíóëÿ. Ïðîòèâîðå÷èå çàâåðøàåò äîêàçàòåëüñòâî.Ñõîäèìîñòü ðàçíîñòíîé çàäà÷è Äèðèõëå75Cëåäñòâèå. Ïóñòü ó íàñ åñòü äâå çàäà÷è:Lh yij = φij ,xij ∈ ωhLh Yij = Φijxij ∈ ωhÏóñòü íà ãðàíèöå âûïîëíÿþòñÿ óñëîâèÿ:yij ≤ Yij ,xij ∈ Γh|φij | ≤ Φij ,xij ∈ ωh|yij | ≤ Yij ,xij ∈ ωhÒîãäà âñþäó âûïîëíåíî:Äîêàçàòåëüñòâî.  ñèëó ëèíåéíîñòè çàäà÷è äëÿ V:Lh Vij = Φij + φijLh ωij = Φij − φijÏðàâûå ÷àñòè îáîèõ óðàâíåíèé íå ìåíüøå íóëÿ â ñèëó âûøåóêàçàííûõ óñëîâèé.

À ýòî, â ñèëóäîêàçàííîãî óòâåðæäåíèÿ, îçíà÷àåò âûïîëíåíèå óñëîâèÿ, êîòîðîå òðåáîâàëîñü äîêàçàòü:|yij | ≤ Yij ,xij ∈ ωhÏåðåïèøåì çàäà÷ó äëÿ ïîãðåøíîñòè àïïðîêñèìàöèè â âèäå:Lh Zij = ψijZij = 0,xij ∈ ωh(2)xij ∈ ΓhÄëÿ äîêàçàòåëüñòâà ñõîäèìîñòè ðàçíîñòíîé ñõåìû íåîáõîäèìî ïîäîáðàòü ìàæîðàíòó÷òîáû âûïîëíÿëîñü óñëîâèå:Lh Yij = K1 , K1 = const > 0Yáóäåì èñêàòü â âèäå:(i) 2(j) 2Yij = (l12 + l22 − (x1 ) − (x2 ) )K,ãäåK>0Yij Lh ≥ 0, xij ∈ ωhYij Lh = 4KÏîëîæèì4K = ||ψ||C :0 = |zij |Γh ≤ Yij |Γh ,4K = ||ψ||C ≥ |ψij |, xij ∈ ωh(|zij | ≤ Yij , zij ∈ ωh )l12 + l22||ψ||C ⇒ ||z||C =≤ M ||ψ||C4ψ = O(h21 + h22 ) ⇒ ||ψ||C ≤ M (l12 + l22 ) ⇒ ||z||C =≤ M 2 (h21 + h22 )||z||C ≤ Yij ≤ (l12 + l22 )K =Òåì ñàìûì, ìû äîêàçàëè ñëåäóþùóþ òåîðåìó:Yòàê,Ìåòîäû ðåøåíèÿ ðàçíîñòíîé çàäà÷è Äèðèõëå76Òåîðåìà 1.

Ïóñòü U (x1 , x2 ) ∈ C 4 (D). Òîãäà ðàçíîñòíàÿ ñõåìà(3) - (4) ñõîäèòñÿ è èìååòìåñòî îöåíêà:||yij − Uij ||C ≤ M1 (h21 + h22 )ÃäåM1íå çàâñèò îòh1èh2 .Äîêàçàòåëüñòâî. Èç ïîëó÷åííîé îöåíêè:||ψ||C ≤ M2 (h21 + h22 )||yij − Uij ||C ≤ M (Ÿ4l12+4l22(3))(h21 + h22 )Ìåòîäû ðåøåíèÿ ðàçíîñòíîé çàäà÷è Äèðèõëåd2 u d2 u+= f (x1 , x2 ) ∈ Ddx21 dx22U |Γu = µ(x1 , x2 )(1)(2)Ðàçðåøèì ñèñòåìó îòíîñèòåëüíî öåíòðàëüíîãî óçëà:(2yi+1,j + yi−1,j yi,j+1 + yi,j−12+ 2 )yij =+− fij2h1 h2h21h22Áóäåì îáîçíà÷àòü èòåðàöèþ ïîä íîìåðîì(s)s − yij(3).Ïðîñòàÿ èòåðàöèÿ (ìåòîä ßêîáè)Ïîëó÷àåì ñëåäóþùèé èòåðàöèîííûé ïðîöåññ:(s)(s)(s)(s)22 (s+1) yi+1,j + yi−1,j yi,j+1 + yi,j−1( 2 + 2 )yij=+− fijh1 h2h21h22s = 0, 1, .

. .(s)yij = µij(0)yij − çàäàíîÄëÿ äîñòèæåíèÿ çàäàííîé òî÷íîñòè òðåáóåòñÿ ïîðÿäêàn0 () ∼ (h−2 ) ∼ (N 2 ), ãäå N = max(N1 , N2 ).Ìåòîä Çåéäåëÿ(s+1)(s)(s+1)(s)22 (s+1) yi−1,j + yi+1,j yi,j−1 + yi,j+1( 2 + 2 )yij+− fij=h1 h2h21h22(s+1)yij= µij , s = 0, 1, . . .ïðè s = 0,yij0 − çàäàíîÏîêàæåì, êàê íàõîäèòü ðåøåíèå: Íà÷íåì ñ óçëà (1, 1), äàëåå äâèæåìñÿ ââåðõ äî (1, n), ïîòîìèç òî÷êè (2, 1) äâèæåìñÿ ââåðõ è ò.ä. Çäåñò áóäåò ðèñóíîê ìåòîäà. Ñ òî÷êè çðåíèÿ îðãàíèçàöèèàëãîðèòìà - íåçíà÷èòåëüíîå óñëîæíåíèå.

Ñ òî÷êè çðåíèÿ ñõîäèìîñòè ìåòîä àíàëîãè÷åí ìåòîäó2ßêîáè: äëÿ ïîëó÷åíèÿ òðåáóåìîé òî÷íîñòè òðåáóåòñÿ ïîðÿäêà n0 () ∼ O(N )Îñíîâíûå ïîíÿòèÿ òåîðèè ðàçíîñòíûõ ñõåì. Àïïðîêñèìàöèÿ. Óñòîé÷èâîñòü. Ñõîäèìîñòü.77Ïîïåðåìåííî-òðåóãîëüíûé èòåðàöèîííûÿ ìåòîä (ìåòîä Ñàìàðñêîãî)Ïåðåïèøåì íàøó ñèñòåìó â âèäå ÑËÀÓ:Ay = φ, ãäå A∗ = A > 0, A = R1 + R20, 5a110......0 ...0, 5a22 0 .