Н.И. Ионкин - Электронные лекции (2009) (1135239), страница 10
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Ââåäåì ñêàëÿðíîå ïðîèçâåäåíèå è íîðìó0â L2 :Z1∀f, g ∈ L2 : (f, g) =f (x)g(x)dx0Z1||f ||L2 = ÂîçüìåìC =√ 21f 2 (x)dx02,ðîâàííûé áàçèñ âòîãäà(uk , ul ) = δkl ,òî åñòüL2 .{uk (x)}∞k=1 - îðòîíîðìèôóíêöèþ f ∈ L2 ìîæíîÒàêèì îáðàçîì, ëþáóþ∞Pïðåäñòàâèòü â âèäå: f (x) =Ck uk (x), ãäå Ck - êîýôôèöèåíòû Ôóðüå.k=1Èìååò ìåñòî ðàâåíñòâî Ïàðñåâàëÿ:||f ||2L2 =∞XCk2k=1Ðàññìîòðèì äèñêðåòíûé àíàëîã çàäà÷è Øòóðìà-Ëèóâèëëÿ: yxx,i + λyi = 0, i = 1, . .
. , N − 1,y0 = yN = 0,yi íå ðàâíû òîæäåñòâåííî 0;(35)Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿòåïëîïðîâîäíîñòè90Ïîäñòàâèì 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 )λ=Èç óñëîâèÿy0 = yN = 04 sin22(1 − cos(αh))=h2h2αh2(36)èìååì:sin α = 0,k∈Zα = πk,Èòàê, ìû íàøëè ñîáñòâåííûå çíà÷åíèÿ è ñîáñòâåííûå ôóíêöèè äèñêðåòíîé çàäà÷è Øòóðìà-Ëèóâèëëÿ: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. Îïðåäåëèì ñêàëÿðíîå ïðîèçâåäåíèå è íîðìó âyk (xi ) =H:∀u, v ∈ H : (u, v) =N−1Xui vi hi=1||u||H =NXi=1! 12u2i hÐàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿòåïëîïðîâîäíîñòè91−1(yk , yl ) = δk l, òîãäà {yk (xi )}Nk=1 - îðòîíîðìèðîâàííûéëþáóþ ôóíêöèþ u ∈ H ìîæíî ïðåäñòàâèòü â âèäå:Ïóñòüu(xi ) =N−1XCk yk (xi ),áàçèñ â H, èi = 0, . . .
, Nk=1Èìååò ìåñòî òàêæå è ðàâåíñòâî Ïàðñåâàëÿ:||u||2H =N−1XCk2k=1Ïóñòüµk (xi ) ≡ yk (xi ),k = 1, . . . , N − 1,i = 0, . . . , N,- ñîáñòâåííûåzin èôóíêöèè äèñêðåòíîé ÇØË (35). Âåðíåìñÿ ê (32) - (34). Ðàçëîæèìψin ïî áàçèñó èç µn :N−1Xzin =k (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 ) =1 − 0, 5τ λkτck (tn ) +ψ (k) (tn )1 + 0, 5τ λk1 + 0, 5τ λkÐàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿòåïëîïðîâîäíîñòèÏîëîæèìqk =921−0,5τ 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−1Xqk ck (tn )µk (xi ) +k=1N−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,(37)èñïîëüçóÿ ðàâåíñòâî Ïàðñåâàëÿ.|qk | < 1 ⇒ kvN−1Xn+1 2k =qk2 c2k (tn )≤k=1N−1Xc2k (tn ) = kz n k2(38)k=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 îòτèïðèh.τ, h → 0,0 < M = const(40)Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿòåïëîïðîâîäíîñòè93Ðàçíîñòíàÿ ñõåìà ñ âåñàìè.
Ïîãðåøíîñòü àïïðîêñèìàöèè.Ïîñòðîèì äëÿ çàäà÷è (1) ðàçíîñòíóþ ñõåìó:yin+1 − yinn+1n+ φni ∈ ωτ h= σyxx,i+ (1 − σ)yxx,iτ(41)y0n+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+1z0n+1 = zN= zi0 = 0Ïîãðåøíîñòü àïïðîêñèìàöèè ðàçíîñòíîé ñõåìû (41) íà ðåøåíèè:un+1− unii+ φniτôóíêöèÿ u(x, t) øåñòün+1ψin = σuxx,i+ (1 − σ)unxx,i −u0 =(42)∂u, u̇ = ∂u.
Ïóñòüðàç íåïðå∂x∂tðûâíî äèôôåðåíöèðóåìà ïî x è òðè ðàçà ïî t. Ðàçëîæèì åå ïî ôîðìóëåÎáîçíà÷èìÒåéëîðà â îêðåñòíîñòè òî÷êè(xi , tn+ 1 ):ui+1 = ui + hu0i +22h 00 h3 000 h4 0000u + ui + ui + . . .2 i624Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿòåïëîïðîâîäíîñòè94ui−1 = ui −hu0ih2 00 h3 000 h4 0000+ ui − ui + ui + . . .2624ττ2τ3111)+u̇(t)+ü(t)+üi (tn+ 1 ) + .
. .un+1=u(ti n+i n+i n+i22222848ττ2τ3uni = ui (tn+ 1 ) − u̇i (tn+ 1 ) + üi (tn+ 1 ) − üi (tn+ 1 ) + . . .22222848uxx,i =ui+1 − 2ui + ui−1h2 000000=u+u + O(h4 )ih212 iun+1i= u̇i (tn+ 1 + O(τ 2 )2τÂîñïîëüçóåìñÿ íåðàâåíñòâîìτ h2 ≤τ 2 +h4:2h2τ+ O(h4 ) + O(τ 2 ))+ψi00 = σ(u00i + u̇00i + u0000212 iτh2(1 − σ)(u00i − u̇00i + u0000+ O(h4 ) + O(τ 2 ))−212 i˙ i + φn + O(τ 2 + h4 ) =(u)ih2 0000u + O(τ 2 + h4 )12 i00Ïðîäèôôèðåíöèðîâàâ óðàâíåíèå u − u̇+f = 0 äâàæäû ïî x, ïîëó÷èì:(u00i − u̇i + φni ) + (σ − 0.5)τ u̇00 +u0000 − u̇00 + f 00 = 0 ⇒ u0000 = u̇00 + f 00Ïîäñòàâèìu0000iâ ôîðìóëó ïîãðåøíîñòè àïïðîêñèìàöèè:h2ψin = u00i − u̇i + f (xi , tn+ 1 ) − f 00 (xi , tn+ 1 ) + φi − f (xi , tn+ 1 )+222|{z} 12=0(σ − 0.5)τ +h2 00u̇ + O(t2 + h4 )12Òàêèì îáðàçîì, ïîðÿäîê ïîãðåøíîñòè àïïðîêñèìàöèè çàâèñèò îò ïàðàìåòðàσè àïïðîêñèìàöèè ôóíêöèè f:Ðàçíîñòíûå ñõåìû äëÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿòåïëîïðîâîäíîñòè1.σ = σ∗ =12−95h212τ⇒φni = f (xi , tn+ 1 ) +2h2 00f (xi , tn+ 1 )212ψin = O(τ 2 + h4 )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 }x26l2ÃÃDÃÃl1x1Ââåäåì 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.Ðàçíîñòíûå ñõåìû äëÿ óðàâíåíèÿ Ïóàññîíà (çàäà÷à Äèðèõëå)96x26Ãl2Ãh2c cc cDh1ÃÃ-l1x1Ãh - ãðàíè÷íûå óçëû.ÃhN1 −1N1 −1N2 −12 −1= {x0,j }Nj=1 ∪ {xN1 ,j }j=1 ∪ {xi,0 }i=1 ∪ {xi,N2 }i=1ωh = ωh ∪ Ãh2Ðàçíîñòíûå 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 },N2 −1N1 −1N1 −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)Ðàçíîñòíûå ñõåìû äëÿ óðàâíåíèÿ Ïóàññîíà (çàäà÷à Äèðèõëå)yij |Γh = µ(xi1 , xj2 ), (xi1 , xj2 ) ∈ Γh97(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 ,i = 1, N1 − 1, j = 1, N2 − 1Ïîãðåøíîñòü ðàçíîñòíîãî ðåøåíèÿZij = 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 ñèñòåìû ñóùåñòâóåò è åäèíñòâåííî. Ïåðåïèøåì ñèñòåìó (3) â âèäå:(2Vi+1,j + Vi−1,j Vi,j+1 + Vi,j−12+ 2 ) ∗ Vi,j =+,2h1 h2h21h22(6)0 < i < N1 ,0 < j < N2 .Òåîðåìà. Ñèñòåìà(6) èìååò òîëüêî òðèâèàëüíîå ðåøåíèå.Äîêàçàòåëüñòâî. Ïðåäïîëîæèì, ÷òî íàéäåòñÿ òàêîé óçåëÒîãäà∃i0 , j0 ,xij , ãäå vij 6= 0.òàêèå ÷òî:à)|Vi0 ,j0 | = max |Vij |0≤i≤N10≤j≤N2b) õîòÿ áû â îäíîì óçëå(i0 , j0 ± 1), (i0 ± 1, j0 )áóäåò âûïîëíåíî|Vi0 ,j0 |Ðàññìîòðèì ðàçíîñòíóþ ñõåìó â óçëå(i0 , j0 :2Vi +1,j + Vi −1,jVi ,j +1 + Vi0 ,j0 −12+ 2 ) ∗ Vi0 ,j0 = 0 0 2 0 0 + 0 02h1 h2h1h22|Vij | <Ñõîäèìîñòü ðàçíîñòíîé çàäà÷è Äèðèõëå98Îöåíèì 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 è3µ.Ñõîäèìîñòü 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 < VijÑõîäèìîñòü ðàçíîñòíîé çàäà÷è Äèðèõëå99Òîãäà: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 ìåíüøå íóëÿ. Ïðîòèâîðå÷èå çàâåðøàåò äîêàçàòåëüñòâî.Cëåäñòâèå.
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