В.Б. Андреев - Численные методы (2 в 1). (2007) (1160465), страница 26
Текст из файла (страница 26)
. , N−1},÷òîmin Ui = Ui0 < 0(19.34)iè â ñèëó (19.27)−(Ui0 −1 − 2Ui0 + Ui0 +1 ) 6 0.210 19. ÝËÅÌÅÍÒÛ ÒÅÎÐÈÈ ÐÀÇÍÎÑÒÍÛÕ ÑÕÅÌÈññëåäóåì îáå ýòè âîçìîæíîñòè. Åñëè −(Ui0 −1 − 2Ui0 + Ui0 +1 ) < 0, òî ñ ó÷åòîì (19.30)è (19.34)Lh1 Ui0 = −Ui0 −1 − 2Ui0 + Ui0 +1 + qi0 Ui0 < 0,è ìû ïðèøëè ê ïðîòèâîðå÷èþ ñ (19.32). Åñëè (Ui0 −1 − 2Ui0 + Ui0 +1 ) = 0, à qi0 6= 0,ìû ñíîâà ïîëó÷àåì ïðîòèâîðå÷èå. Äëÿ âûõîäà èç ýòèõ ïðîòèâîðå÷èé ìû äîëæíûïðåäïîëîæèòü, ÷òî qi0 = 0 è (Ui0 −1 − 2Ui0 + Ui0 +1 ) = 0. Íî â ñèëó (19.27), (19.34)ýòî îçíà÷àåò, ÷òî Ui0 −1 = Ui0 = Ui0 +1 < 0, è â êà÷åñòâå i0 èç (19.34) ìîæíî âçÿòüòàêæå (i0 − 1) èëè (i0 + 1).
Äåëàÿ ýòîò âûáîð, ìû òåìè æå ðàññóæäåíèÿìè ïðèõîäèìê óòâåðæäåíèþ, ÷òî è Ui0 −2 = Ui0 (èëè Ui0 +2 = Ui0 ). È ò.ä. Ïîñêîëüêó â ñèëó (19.31),(19.34) ôóíêöèÿ Ui , i = 0, N íå ÿâëÿåòñÿ ïîñòîÿííîé, òî ñóùåñòâóåò òàêîé óçåë xi1 ,i1 ∈ {1, 2, . . . , N − 1}, ÷òî Ui1 = Ui0 , à Ui1 −1 èëè Ui1 +1 áîëüøå Ui1 .  ýòîì óçëå −(Ui1 −1 −2Ui1 + Ui1 +1 ) < 0, è ìû âåðíóëèñü ê óæå ðàññìîòðåííîìó ñëó÷àþ, êîòîðûé ïðèâåëíàñ ê ïðîòèâîðå÷èþ ñ (19.32). Âñå ïðîòèâîðå÷èÿ ñíèìàþòñÿ, åñëè ìû îòêàæåìñÿ îòïðåäïîëîæåíèÿ, ÷òî Ui ìîæåò ïðèíèìàòü îòðèöàòåëüíûå çíà÷åíèÿ.
Òåîðåìà äîêàçàíà.Îïðåäåëåíèå 19.10. Ìàòðèöà A íàçûâàåòñÿ ìîíîòîííîé, åñëè ëþáîé âåêòîð x, äëÿêîòîðîãî Ax > 0, ÿâëÿåòñÿ íåîòðèöàòåëüíûì.Òåîðåìà 19.4 (Ïðèíöèï ñðàâíåíèÿ). Ïóñòü uhi ðåøåíèå çàäà÷è (19.22), (19.9),à Ui ðåøåíèå ñëåäóþùåé çàäà÷è:Lh1 Ui = Fi ,i = 1, N − 1,U0 = G0 ,UN = G1 .Ïóñòü|fi | 6 Fi ,|g0 | 6 G0 ,|g1 | 6 G1 .(19.35)Òîãäà, åñëè âûïîëíåíî óñëîâèå (19.30), òî|uhi | 6 Ui ,i = 1, N − 1.(19.36)Äîêàçàòåëüñòâî. Ëåãêî âèäåòü, ÷òî ôóíêöèÿ (Ui − uhi ) ÿâëÿåòñÿ ðåøåíèåì çàäà÷èLh1 (U − uh )i = Fi − fi ,i = 1, N − 1,U0 − uh0 = G0 − g0 ,UN − uhN = G1 − g1 . ñèëó (19.35) è òåîðåìû 19.3 çàêëþ÷àåì, ÷òî Ui − uhi > 0. Èç àíàëîãè÷íûõ ñîîáðàæåíèé íàõîäèì, ÷òî è Ui + uhi > 0. Òåì ñàìûì, −Ui 6 uhi 6 Ui , è òåîðåìà äîêàçàíà.Çàìå÷àíèå 19.4. Ôóíêöèÿ Ui èç (19.36) íàçûâàåòñÿ áàðüåðîì.Òåîðåìà 19.5. Äëÿ ðåøåíèÿ çàäà÷è (19.22), (19.9) ïðè âûïîëíåíèè óñëîâèÿ (19.30)ñïðàâåäëèâà àïðèîðíàÿ îöåíêàmax |uhi | 6 |g0 | + |g1 | +il2max |fi |.8 i19.4.
ÓÐÀÂÍÅÍÈß Ñ ÏÅÐÅÌÅÍÍÛÌÈ ÊÎÝÔÔÈÖÈÅÍÒÀÌÈ211Äîêàçàòåëüñòâî. Ââåäåì â ðàññìîòðåíèå ôóíêöèþ(19.37)Ui = |g0 |(1 − xi ) + xi |g1 | + c xi (1 − xi ) > 0,ãäå c > 0 íåêîòîðàÿ ïîñòîÿííàÿ. Î÷åâèäíî, ÷òî U0 = |uh0 |, UN = |uhN |. Ëåãêîïðîâåðèòü, ÷òîLh1 Ui = 2c + qi Ui =: Fi > 2c.Ïóñòü c = 1/2 max |fi |. Òîãäà |fi | 6 Fi , è ìû íàõîäèìñÿ â óñëîâèÿõ òåîðåìû 19.4, ò.å.|uhi | 6 Ui . Íîimax Ui 6 |g0 | + |g1 | + c/4.iÒåîðåìà äîêàçàíà.Óïðàæíåíèå 19.2.
Ñôîðìóëèðîâàòü è äîêàçàòü òåîðåìó î ñêîðîñòè ñõîäèìîñòèðàçíîñòíîé çàäà÷è (19.22), (19.9).19.4 Óðàâíåíèÿ ñ ïåðåìåííûìè êîýôôèöèåíòàìèÐàññìîòðèì îáùåå ñàìîñîïðÿæåííîå óðàâíåíèå âòîðîãî ïîðÿäê൶ddu−p(x)+ q(x)u = f (x),0 < x < 1.dxdx(19.38)è èçó÷èì âîïðîñ î åãî àïïðîêñèìàöèè. Íà ïåðâûé âçãëÿä êàæåòñÿ âïîëíå åñòåñòâåííûì ðàçäèôôåðåíöèðîâàòü ïåðâîå ñëàãàåìîå ëåâîé ÷àñòè (19.38)−p(x)dud2 u0−p(x)+ q(x)u = f (x)d x2dx(19.39)è â ýòîì âèäå çàìåíèòü d2 u/dx2 è du/dx ñîîòâåòñòâóþùèìè ðàçíîñòíûìè îòíîøåíèÿìè. Íî òàê ïîñòóïàòü ïëîõî â ñèëó öåëîãî ðÿäà ïðè÷èí.  ÷àñòíîñòè, óðàâíåíèå (19.38)ÿâëÿåòñÿ ôîðìàëüíî ñàìîñîïðÿæåííûì ïî Ëàãðàíæó (ñèììåòðè÷íûì, ò.å. åñëè Lv :=R1R1−(pv 0 )0 +qv , à u(x) è v(x) îáðàùàþòñÿ â íóëü ïðè x = 0 è x = 1, òî vLu d x = uLv d x.00Ñðàâíèòü ñ ñèììåòðè÷íîé ìàòðèöåé A = AT (Ax, y) = (x, Ay)).
Åñëè æå àïïðîêñèìèðîâàòü (19.39), êîòîðîå ýêâèâàëåíòíî (19.38) ïðè ãëàäêîé p(x), òî àïïðîêñèìàöèÿ,âîîáùå ãîâîðÿ, ñèììåòðè÷íîé íå áóäåò. Óðàâíåíèå (19.38) íóæíî àïïðîêñèìèðîâàòüñðàçó â èñõîäíîì âèäå.Ïîñòðîèì àïïðîêñèìàöèþ (19.38) ïðè ïîìîùè èíòåãðî - èíòåðïîëÿöèîííîãî ìåòîäà (ìåòîäà áàëàíñà, ìåòîäà êîíå÷íûõ îáúåìîâ). Ïóñòü xi±1/2 = xi ± h/2. Ïðîèíòåãðèðóåì óðàâíåíèå (19.38) ïî îòðåçêó (xi−1/2 , xi+1/2 ).
Áóäåì èìåòü− p(xi+1/2 )u0 (xi+1/2 ) + p(xi−1/2 )u0 (xi−1/2 )+Z xi+1/2+[q(x)u(x) − f (x)] d x = 0.xi−1/2(19.40)212 19. ÝËÅÌÅÍÒÛ ÒÅÎÐÈÈ ÐÀÇÍÎÑÒÍÛÕ ÑÕÅÌÇàìåíèì â (19.40) èíòåãðàë êâàäðàòóðíîé ôîðìóëîé ïðÿìîóãîëüíèêîâ, à ïðîèçâîäíûå ñîîòâåòñòâóþùèìè ðàçíîñòíûìè îòíîøåíèÿìè. Èìåííî×xi−1/2xi−1xi×xi+1/2xi+1Ðèñ. 1xi+1/2Z[q(x)u − f (x)] dx ≈ qi ui h − fi h,(19.41)xi−1/2ui − ui−1ui+1 − ui, u0i−1/2 ≈.hhÏîäñòàâëÿÿ (19.41) â (19.40), ïîëó÷èì ïðèáëèæåííîå ðàâåíñòâî.
Çàìåíÿÿ ïðèáëèæåííîå ðàâåíñòâî íà òî÷íîå, ïîëó÷èì óðàâíåíèå äëÿ ïðèáëèæåííîãî ðåøåíèÿ. Ïîñëåäåëåíèÿ íà h îíî ïðèìåò âèä:·¸uhi+1 − uhiuhi − uhi−11− pi+1/2− pi−1/2+ qi uhi = fi , i = 1, N − 1(19.42)hhhu0i+1/2 ≈Ââåäåì ñëåäóþùèå îáîçíà÷åíèÿui+1 − ui ïðàâîå ðàçíîñòíîå îòíîøåíèå,hui − ui−1 ëåâîå ðàçíîñòíîå îòíîøåíèå.ux̄ := ux̄,i :=hÎ÷åâèäíî, ÷òî vx,i ≡ vx̄,i+1 . Äàëåå,·¸vi+1 − 2vi + vi−11 vi+1 − vi vi − vi−1=−=h2hhh11= (vx,i − vx̄,i ) = (vx̄,i+1 − vx̄,i ) = (vx̄ )x,i =hh= vx̄x,i =: vx̄x .ux := ux,i :=Èñïîëüçóÿ ââåäåííûå îáîçíà÷åíèÿ, óðàâíåíèå (19.42) ìîæíî ïåðåïèñàòü òàê:¡¢(19.43)− ph uhx̄ x,i + qih uhi = fih , i = 1, N − 1,ãäåµhp :=phih:= p xi −2¶,q h := qih := q(xi ),f h := fih := f (xi ).(19.44)19.5.
ÀÏÏÐÎÊÑÈÌÀÖÈß ÃÐÀÍÈ×ÍÛÕ ÓÑËÎÂÈÉ21319.5 Àïïðîêñèìàöèÿ ãðàíè÷íûõ óñëîâèéÏðèìåíèì òåïåðü èíòåãðî-èíòåðïîëÿöèîííûé ìåòîä äëÿ ïîñòðîåíèÿ àïïðîêñèìàöèèãðàíè÷íîãî óñëîâèÿ, ñîäåðæàùåãî ïðîèçâîäíóþ. Ïóñòü äëÿ óðàâíåíèÿ (19.38) â òî÷êåx = 0 (ãðàíè÷íîé òî÷êå) çàäàíî ãðàíè÷íîå óñëîâèåαdu(0)+ βu(0) = γ.dx(19.45)Ãðàíè÷íîå óñëîâèå (19.45) ñîäåðæèò â ñåáå âñå îñíîâíûå ãðàíè÷íûå óñëîâèÿ äëÿóðàâíåíèÿ (19.38): èìåííî, ãðàíè÷íûå óñëîâèÿ ïåðâîãî ðîäà (α = 0), âòîðîãî ðîäà(β = 0) è òðåòüåãî ðîäà. Íàñ áóäóò èíòåðåñîâàòü ãðàíè÷íûå óñëîâèÿ âòîðîãî è òðåòüåãîðîäà, ò.å. óñëîâèÿ, ñîäåðæàùèå ïðîèçâîäíóþ.
Ïðîñòåéøàÿ àïïðîêñèìàöèÿ óñëîâèÿ(19.45) èìååò âèäuh − uh0α 1+ βuh0 = γ.(19.46)hÓïðàæíåíèå 19.3. Äîêàçàòü, ÷òî ïîãðåøíîñòü àïïðîêñèìàöèè ãðàíè÷íîãî óñëîâèÿ(19.45) ãðàíè÷íûì óñëîâèåì (19.46) ïðè α 6= 0 åñòü O(h).Ìû íå áóäåì çàíèìàòüñÿ ýòîé àïïðîêñèìàöèåé èç-çà òîãî, ÷òî îíà èìååò áîëüøóþ ïîãðåøíîñòü. Ïîñòðîèì äðóãóþ àïïðîêñèìàöèþ óñëîâèÿ (19.45), íî ïðåæäå åãîíåñêîëüêî ïðåîáðàçóåì.
Ïî ïðåäïîëîæåíèþ α 6= 0, è íà ýòîò êîýôôèöèåíò óñëîâèå(19.45) ìîæíî ðàçäåëèòü. Êîýôôèöèåíò p(x) óðàâíåíèÿ (19.38) áóäåì ïðåäïîëàãàòüñòðîãî ïîëîæèòåëüíûìp(x) > c0 > 0,(19.47)è äîìíîæåíèå (19.45) íà −p(0) ïðèâåäåò ê ýêâèâàëåíòíîìó óðàâíåíèþ. Áóäåì âìåñòî(19.45) ðàññìàòðèâàòü ãðàíè÷íîå óñëîâèå−p(0)du(0)+ κ0 u(0) = g0 ,dx(19.48)êîòîðîå ïðè α = −p(0) 6= 0, β = κ0 è γ = g0 ñîâïàäàåò ñ (19.45). Êîìáèíàöèÿ p(0)u0 (0)â (19.48) õîðîøà óæå òåì, ÷òî âåëè÷èíà −p(x)u0 (x) èìååò ñìûñë ïîòîêà è ôèãóðèðóåò âñàìîì óðàâíåíèè (19.38). Çíàê ìèíóñ ïåðåä ïðîèçâîäíîé äîëæåí ñâèäåòåëüñòâîâàòü îòîì, ÷òî ïðîèçâîäíàÿ áåðåòñÿ ïî "âíåøíåé íîðìàëè": ïðîèçâîäíàÿ du(0)/dx âû÷èñëåíà ïî íàïðàâëåíèþ âíóòðü îòðåçêà [0, 1], à ïðîèçâîäíàÿ −du(0)/dx ïî íàïðàâëåíèþ,âûõîäÿùåìó èç îòðåçêà.×òîáû ïîñòðîèòü àïïðîêñèìàöèþ (19.48), ïðîèíòåãðèðóåì óðàâíåíèå (19.38) ïîîòðåçêó (0, h/2).
Áóäåì èìåòüdu(h/2)du(0)−p(h/2)+ p(0)+dxdxZh/2[q(x)u(x) − f (x)] dx = 0.0(19.49)214 19. ÝËÅÌÅÍÒÛ ÒÅÎÐÈÈ ÐÀÇÍÎÑÒÍÛÕ ÑÕÅÌÇàòåì âûðàçèì p(0)du(0)/dx èç (19.48)p(0)du(0)= κ0 u(0) − g0 ,dx(19.50)àïïðîêñèìèðóåì ïðîèçâîäíóþdu(h/2)u1 − u 0≈(19.51)dxhè àïïðîêñèìèðóåì èíòåãðàë â (19.49) êâàäðàòóðíîé ôîðìóëîé "ëåâûõ ïðÿìîóãîëüíèêîâ"Zh/2h[q(x)u(x) − f (x)] dx ≈ [q(0)u(0) − f (0)] .(19.52)20Ïîäñòàâëÿÿ òåïåðü (19.50)-(19.52) â (19.49), ïîëó÷èì ïðèáëèæåííîå ðàâåíñòâî, êîòîðîå ïðåâðàòèì â òî÷íîå ïóòåì çàìåíû òî÷íîãî ðåøåíèÿ u(x) íà ïðèáëèæåííîå uh (x).Áóäåì èìåòüµ¶uh1 − uh0hh−p1/2+ κ0 + q0 uh0 = g0 + f0h22èëè, ïðèíèìàÿ îáîçíà÷åíèÿ (19.44),hh−ph1 uhx̄,1 + (κ0 + q0h )uh0 = g0 + f0h .22Ñîîòíîøåíèå (19.53) ïðåäñòàâëÿåò ñîáîé èñêîìóþ àïïðîêñèìàöèþ.(19.53)19.6 Èññëåäîâàíèå ïîãðåøíîñòè àïïðîêñèìàöèèÈññëåäóåì ïîãðåøíîñòü àïïðîêñèìàöèè ðàçíîñòíîé ñõåìû (19.43). Èññëåäóåì äàæåáîëåå îáùóþ ñõåìó.
Ïóñòü ðàçíîñòíàÿ ñõåìà èìååò âèä¤1£− bi uhx,i − ai uhx̄,i + qih uhi = fih .(19.54)hÏîãðåøíîñòü àïïðîêñèìàöèè ýòîé ñõåìû åñòü1[bi ux,i − ai ux̄,i ] − qih ui =h= [fih − f (xi )] − [qih − q(xi )]ui +1+ [bi ux,i − ai ux̄,i ] − (pu0 )0i .hΨi = fih +Ïðè u(x) ∈ C 4 [0, 1] èìåþò ìåñòî ñëåäóþùèå ðàçëîæåíèÿhux,i = u0i + u00i +2hux̄.i = u0i − u00i +2h2 000u + O(h3 ),6 ih2 000ui + O(h3 ).6(19.55)19.6. ÈÑÑËÅÄÎÂÀÍÈÅ ÏÎÃÐÅØÍÎÑÒÈ ÀÏÏÐÎÊÑÈÌÀÖÈÈ215Ïîäñòàâëÿÿ ýòè ñîîòíîøåíèÿ â (19.55), áóäåì èìåòü·1hh23Ψi =bi (u0i + u00i + u000i + O(h ))−h26¸2h 00 h 00003− ai (ui − ui + ui + O(h )) −260 000− (p u + pu )−− [qih − q(xi )]ui + [fih − f (xi )] =µµ¶¶bi − aibi + ai00=− p i ui +− pi u00i +h2bi − ai 000+hui + O(h2 ) − (qih − qi )ui +6+ (fih − fi ).Îòñþäà íàõîäèì, ÷òî äëÿ àïïðîêñèìàöèè O(h2 ) íåîáõîäèìî è äîñòàòî÷íî âûïîëíåíèÿóñëîâèébi − ai1◦ .− p0i = O(h2 ),hbi + ai2◦ .− pi = O(h2 ),(19.56)23◦ .qih − qi = O(h2 ),4◦ .fih − fi = O(h2 ).Äëÿ ñõåìû (19.43), (19.44) óñëîâèÿ (19.563 ) è (19.564 ) î÷åâèäíû.
Îáðàòèìñÿ ê (19.561 )è (19.562 ). Èìååìhh2bi = pi+1/2 = pi + p0i + p00i + O(h3 ),28h 0 h2 00ai = pi−1/2 = pi − pi + pi + O(h3 ).28Îòñþäàbi − ai= p0i + O(h2 ),hbi + a i= pi + O(h2 ).2Òåîðåìà 19.6. Åñëè ðåøåíèå óðàâíåíèÿ (19.38) îáëàäàåò ÷åòâåðòûìè íåïðåðûâíû-ìè ïðîèçâîäíûìè, òî ðàçíîñòíàÿ ñõåìà (19.43), (19.44) èìååò ïîãðåøíîñòü àïïðîêñèìàöèè O(h2 ).Óïðàæíåíèå 19.4.
Äîêàçàòü, ÷òî ðàçíîñòíàÿ ñõåìà (19.43) ïðè bi = ai+1 èà)ai =pi + pi−1,2qih = qi ,fih = fi ,(19.57)216 19. ÝËÅÌÅÍÒÛ ÒÅÎÐÈÈ ÐÀÇÍÎÑÒÍÛÕ ÑÕÅÌZxi1á) ai =hp(x) dx,qihxi−1xi+1Zfih =1hxi+1Zq(x)(1 − |x − xi |) dx,1=hxi−1(19.58)f (x)(1 − |x − xi |) dxxi−1èìååò ïîãðåøíîñòü àïïðîêñèìàöèè O(h2 ).Èññëåäóåì ïîãðåøíîñòü àïïðîêñèìàöèè ψ0 ãðàíè÷íîãî óñëîâèÿ (19.53). Èìååìhhu1 − u0ψ0 := g0 + f0 + p1/2− (κ0 + q0 )u0 =2 µh¶µ 2¶h 00hh 0h202= g0 + f0 + p0 + p0 + O(h )u0 + u0 + O(h ) − (κ0 + q0 )u0 =2222h= (p0 u00 − κ0 u0 + g0 ) + (p0 u000 + p00 u00 − q0 u0 + f0 ) + O(h2 ).2Ïåðâàÿ ñêîáêà â ýòîì ïðåäñòàâëåíèè ðàâíà íóëþ â ñèëó (19.48), à âòîðàÿ â ñèëóóðàâíåíèÿ (19.38), ïðîäîëæåííîãî ïî íåïðåðûâíîñòè ñ (0, 1) íà [0, 1).
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
