И.О. Арушунян - Конспект лекций (1160471), страница 10
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(ò.ê. íîñèòåëè ó ϕk è ϕk+2 , ϕk+3 , ... íå ïåðåñåê.- ñì. Aij :=)Äëÿ âû÷èñëåíèÿ êîýô. ïðèìåíÿþòñÿ êâàäð. ô-ëû.Ïðèìåð:½Z 1Z 1−y 00 (x) = f (x)0⇒ Aij =ϕi (x)ϕj (x)dx, bk =f (x)ϕk (x)dxy(0) = y(1) = 000Z1. i = j : Aii =xi +hxi −hZ2. i 6= j : Ai i+1 =(ϕ0i )2 dx =xi +hxi2hµϕ0i ϕ0i+1 dxA=1=h − 2h2h− h10− h1¶02h− h1...............···− h1820− h12h−yi−1 + 2yi − yi+1, òî ïîëó÷àëè òàêóþ æå ìàòðèöó, íî âh2R2çíàìåíàòåëÿõ ñòîÿëè íå h, à h . (ðàçíèöà â òîì, ÷òî ñïðàâà ñòîÿò bk = ...)∗ Êîãäà çàìåíÿëè −y 00 =ZZxk +hbk =xk +hf (x)ϕk (x)dx ≈ f (xk )ϕk (x)dx = hf (xk ) ⇒xk −hxk −hïîëó÷èëè òàêóþ æå ñèñòåìó, êàê è ïðè çàìåíå (ñì. âûøå)Çàäà÷à Áóáíîâà-ÃàëåðêèíàÁîëåå îáùàÿ ïîñòàíîâêà â îòëè÷èå îò ìåòîäà ÐèòöàyN (x) =NP−1cj ϕj (x)j=0qRb 2ky − yN kL2 = O(h ), ãäå kukL2 =u dxqRab 2(u + (u0 )2 )dxky − yN kW21 = O(h), ãäå kukW21 =a2½NPLy = 0cj ϕj (x), ãäå Ly = −(ky 0 )0 + py − f, yN =y(a) = y(b) = 01Âîçüìåì åùå îäèí íàáîð ë.í.ô. ψ1 , ..., ψN ∈ W21RbÏîòðåáóåì (LyN , ψk ) = 0, k = 1, ..., N , ãäå (f, g) = a f (x)g(x)dx ⇒ îïðåäåëÿåìêîýô-òû ðàçë-ÿ.Ïîòðåáóåì k(x) ≥ k0 > 0, p ≥ 0ZbaZbaÏîäñòàâèì yN =Z0 0(−(kyN) ϕk (x))dxNP=0−(kyN(x))ϕk (x)|baq0b+a0kyNϕ0k dxci ϕi (x)i=1N ·ZXi=10 0(−(kyN) ϕk (x) + pyN ϕk (x) − f (x)ϕk (x))dx = 0ba¸(k(x)ϕ0i (x)ϕ0k (x))dxZb=f (x)ϕk (x)dx, k = 1, ..., NaÓ ìåòîäà Áóáíîâà-Ãàëåðêèíà L - îáùåãî âèäà.µ¶∂u∂u∂u∂+a= f (x, t),+(a(x, t)u(x)) = f (x, t), − îáîáùåíèå∂t∂x∂t∂x83t6îáëàñòü çàäà÷è-x∂u∂ 2uÓðàâíåíèå òåïëîïðîâîäíîñòè= a2 2 = f (x, t)∂t∂x½−∆u = f (x, y), (x, y) ∈ Ω− çàäà÷à Äèðèõëå äëÿ îïåð-ðà Ëàïëàñàu|∂Ω = ϕ(x, y)µ¶2P∂∂uÎáîáùåíèåkij= f (x1 , x2 ).∂xji,j=1 ∂xiÎáùèå îáîçíà÷åíèÿ:∂Ω =ΩsSΓii=1.
äèôô.îïåð-ðÇàäà÷à: Lu = f, x ∈ Ωli u = ϕi , x ∈ Γ iτΩh ⊂ Ω̄ &% ìí-âà òî÷åêΓi ∼ ΓhÌí-âî ô-èé, îïðåäåë. â Ωh - Uh .L h uh = f hx ∈ Ωhlih = ϕhi[u]h - ñåò. ô-ÿ, ñîâï. ñ ñàìî́é íåïð. ô-åé â óçëàõ ñåòêè.k[u]h kUh → kukU - óñë-å ñîãëàñîâàííîñòè íîðì.hh→0Àïïðîêñèìàöèÿ, óñòîé÷èâîñòü, ñõîäèìîñòü.Îïðåäåëåíèå. Ðàçíîñòíàÿ çàäà÷à àïïðîêñèìèðóåò äèôô. çàäà÷ó, åñëèkLh [u]h − fh kUh +sXklih [u]h − ϕhi kΦhi → 0h→0i=1Îïðåäåëåíèå.
Ðåøåíèå ðàçí. çàä. ,̈_ ñõîäèòñÿ ê ðåø-þ äèôô. çàäà÷è, åñëèk[u]h − uh kUh → 0h→084Îïðåäåëåíèå. Óñòîé÷èâîñòü:kuh kUh ≤ C(kfh kFh +Xkϕhi kΦhi )Àïïð.+Óñò-òü ⇒ Ñõ-òü.Êîíñïåêòû ëåêöèé íå ðåäàêòèðîâàëèñü! Âíå âñÿêîãî ñîìíåíèÿ íà ëåêöèè ïðîçâó÷àëèâñå íåîáõîäèìûå ñëîâà, êîòîðûå ïðåâðàùàþò â êîððåêòíûå îïðåäåëåíèÿ òî, ÷òîíàïèñàíî âûøå. Åñëè ïðè îòâåòå íà ýêçàìåíå äàòü òàêèå îïðåäåëåíèÿ, êàê â ýòîìêîíñïåêòå - ýòî ãàðàíòèðîâàííûé "íåóä"!!!!21Ëåêöèÿ 21(8).Lu = f,S x∈Ω∂Ω = Γiili u = ϕi , x ∈ ΓiÐàçíîñòíàÿ çàäà÷à:L h uh = f hlih uh = ϕihõîòèì: Ñõîäèìîñòü:k[u]h − uh kUh → 0 (h → 0)[u]h - ðåøåíèå, âçÿòîå â óçëàõ ñåòêè.Àïïðîêñèìàöèÿ:kLh [u]h − fh kFh +Xklih [u]h − ϕh kΦh → 0iÓñòîé÷èâîñòü: (äëÿ ëèíåéíîãî ñëó÷àÿ)Ãîâîðÿò, ÷òî ð.ç.
óñòîé÷èâà, åñëè ∃ C íåçàâ. îò h : ∀ ñåò. ô. fh , ϕihXkuh kUh ≤ C(k Lh uh kFh +k lih uh kΦh )| {z }|{z}ifhϕih ëèí. ñëó÷àå: åñëè çàäà÷à óñòîé÷èâà, òî ðåøåíèå ∃ è ! (fh = 0, ϕih = 0 ⇒ uh = 0)Ôîðìàëüíî: Lh uh = fh ⇒ íîðìà îïåðàòîðà Lh äîëæíà áûòü îãðàíè÷åíà.Ãäå-òî ïîòåðÿëèñü ñëîâà ïðî ñîãëàñîâàííîñòü íîðì!Òåîðåìà Ôèëèïïîâà: èç àïïðîêñèìàöèè è óñòîé÷èâîñòè ñëåäóåò ñõîäèìîñòü.Äîê-âî:Lh uh = fhLh [u]h + αh , kαh k → 0ìàëîlih uh = ϕih , lih [u]h = ϕh + αh , kαhi k → 0zh = [u]h − uh − ðåøåíèå çàäà÷èLh zh = αh , lih zh = αhi85- ýòî èç àïïðîêñèìàöèè.Èç óñòîé÷èâîñòè:kzh kUh ≤ C(kαh k +Piqk[u]h − uh kUh .Ðàññì.
çàäà÷ó:(kαhi k) → 0∂u ∂u−= f (x, t), t ≥ 0∂t∂xu(x, 0) = ϕ(x)Ñåòêà: ïî x: øàã h, ïî t: øàã τ .Ñåòî÷í. ô-ÿ: uh (mh, nτ ) = unm , m ∈ Z, n ∈ N0nunm+1 − unm un+1m − umn−= fmτh u0 = ϕmmu(mh, (n + 1)τ ) − u(mh, nτ ) u((m + 1)h, nτ ) − u(mh, n)−− f (mh, nτ ) =ττu(., .) ðàñêëàäûâ. â ðÿä Òåéëîðà:∂u∂u+ O(τ ) −+ O(h) − f (x, t) = O(τ + h)∂t∂x(÷àùå âñåãî τ = const · h)⇒ àïïðîêñèìàöèÿ åñòü.τÓñòîé÷èâîñòü íå âñåãäà. Îáîçíà÷èì = qhÒåîðåìà: q ≤ 1 ⇒ ñõåìà óñòîé÷èâàÄ-âî: kuh k = sup |unm |.
Íàäî äîêàçàòü=m,nnsup |unm | ≤ C(sup |fm| + sup |ϕm |).m,nm,nmnun+1= (1 − q)unm + qunm+1 + τ fmmnnn|un+1m | ≤ |(1 − q)um + qum+1 | + |τ fm | ≤nn≤ sup |um ||(1 − q) + q| + τ sup |fm | ≤m,nmn|≤ sup |unm | + τ sup |fmm,nmÏóñòü t ≤ T(íà ïðàêòèêå îáû÷íî íóæíî âû÷-òü ðåøåíèå íà êîíå÷íîì îòðåçêå t ∈ [0, T ])T = Nτnn|un+1m | ≤ sup |um | + τ sup |fm | ≤ ...m,nmn|=...
≤ sup |u0m | + N τ sup |fmm,nmkϕk + T · kf k ≤ T (kϕk + kf k).86Åñëè q = τ /h > 1 ⇒ çàäà÷à íå óñòîé÷èâà, äîñò. ä-òü, ÷òî íåò ñõ-òè.Ñ÷èòàåì f ≡ 0nun − unm un+1m − um− m+1=0h u0 =τ ϕmmt6t6(0,1) •••••••••••••••?(1,0)-(0,1)•@Ëèíèÿ (0,1)(1,0)• •@õàðàêòåðèñòèêà äèôô. çàäà÷è@•••@•••• @• • • • • @?(1,0)xxÄëÿ âû÷èñëåíèÿ ðåøåíèÿ ðàçíîñòíîé ñõåìû â êðàñíûõ ïóëÿõ òðåáóþòñÿ çíà÷åíèÿâ ÷åðíûõ ïóëÿõ íà íóëåâîì ñëîå ïî t. Äëÿ âû÷-ÿ â çåëåíûõ ïóëÿõ - òðåá. çíà÷åíèÿ âêðàñíûõ. Äëÿ ñèíèõ - òðåáóþòñÿ â çåëåíûõ ... Äëÿ âû÷èñëåíèÿ â òî÷êå (0,1) òðåáóþòñÿçíà÷åíèÿ â íàðèñîâàííûõ ïóëÿõ.Ðåøåíèå äèôô. çàäà÷è ïîñòîÿííî âäîëü ïðÿìûõ x−t = const - õàðàêòåðèñòèê.
Åñëèèçìåíèì íà÷àëüíîå óñëîâèå çàäà÷è â òî÷êå (1,0), òî ðåøåíèå äèôô. çàäà÷è èçìåíèòñÿâ òî÷êå (0,1), à ðåøåíèå ðàçíîñòíîé ñõåìû ïðè τ > h â ýòîé òî÷êå íå èçìåíèòñÿ, çíà÷èòñõîäèìîñòè áûòü íå ìîæåò.Ñïåêòðàëüíûé ïðèçíàê óñòîé÷èâîñòè.(Ïðèçíàê óñòîé÷èâîñòè ïî íà÷àëüíûì äàííûì.)Åñëè çàäà÷à óñòîé÷èâà, òî îíà óñòîé÷èâà ïðè u0m = eimα .Ðåøåíèåì ðàçíîñòíîé ñõåìû ñ òàêèì u0m áóäåò unm = λn eimα , λ = λ(α) íàõîäèì èçïîäñòàíîâêè unm â ð. ñõåìóλn+1 eimα = λn eimα + qλn (ei(m+1)α − eimα )λ = 1 + q(eiα − 1)α ∈ [0, 2π]Óñòîé÷èâîñòü â äàííîì ñëó÷àå: îãðàíè÷åííîñòü kunm k1.
|λ| ≤ 1 - ñòðîãîå óñëîâèå2. |λ| ≤ 1 + C1 τ - óñëîâíàÿ óñò-òü (äîñò. äëÿ îãð-òè kuk)2. èñïîëüçóåòñÿ, åñëè ñ÷èòàåì çàäà÷ó â ïîëîñå [0, T ], ⇒ êîë-âî øàãîâ îãð-íî: n ≤T /τ ⇒|unm | ≤ |λ|n ≤ (1 + C1 τ )n ≤ eC1 T ;åñëè ýòî óñëîâèå íå âûï-ñÿ, òî ñõåìà íåóñòîé÷èâà.Äðóãèå ðàçíîñòíûå ñõåìû:•nun − unm−1un+1m − umn− m= fmτh87•nun+1 − un+1un+1mm − umn− m+1= fmτhËó÷øàÿ âûáèð-ñÿ ñ ïîì. ïîäñò. ðåø.
u0m = eimα , unm = λn eimα è ðàññì-ÿ ñîîòâ. λ.Ïóñòü íóæíî ðåø. â êîíå÷íîì ïðÿìîóã-êå [a, b] × [0, T ]t6Tnun − unmun+1m − umn− m+1= fmτhllòðåáóåòñÿ íàéòè çíà÷åíèÿ íà [b, (b, T )]ll⇒ íóæíû çíà÷åíèÿ íà [b, c]lò.å. íà ðàñøèð. îòð. [a, c]ll [b, c] îïð-ñÿ êîë-âîì øàãîâ è T .bc x(b, T )a,,Äëÿ ñõåìû∂∂u−(a, u) = f − òàêàÿ çàäà÷à∂t∂xëèíåàðèçóåòñÿ.,Óðàâíåíèå òåïëîïðîâîäíîñòè:t6Tðàñ÷. îáë.τ ðàâí. ñåò.ñ øàã. τ, h2∂u∂ u= a2 2 + f (x, t)∂t∂xu(x, 0) = ϕ(x), x ∈ [0, T ]u(0, t) = µ1 (t)u(X, t) = µ2 (t)-0ßâíàÿ ñõåìà:hXxnun − 2unm + unm−1un+1m − umn= a2 m+1+ fmτh2! Ïîð. àïïðîêñèìàöèè: O(τ + h2 )Óñòîé÷èâîñòü ïî íà÷. äàííûì ðàññì. äëÿ çàä.:t6 ∂u∂ 2u= a2 2∂t∂x u(x,0) = ϕ(x)îáëàñòü çàäà÷è← (áîëåå îáùàÿ çàäà÷à) ??x-88unm = λn eimαλ−1eiα − 2 + e−iα− a2=0τh2 µ¶2eiα − 2 + e−iαeiα/2 − e−iα/2=−= − sin2 α/242iq = τ /h2 ⇒λ(α) = 1 − 4qa2 sin2 α/2 ⇒ λ ∈ Rλ ∈ [1 − 4qa2 , 1]1|λ| ≤ 1 ⇒ q ≤ 22aÍà ïðàêòèêå ðåø. îáû÷íî áîëåå ãëàäêî ïî t, ÷åì ïî x è ïîëåçíî τ = const · h, à íåτ ∼ h2 .
• n + 1]• n- øàáëîí ñõåìû.•Á•61q ≤ 2 - çàìåäëÿåò ñ÷åò, ïîýòîìó ëó÷øå èñï-òü íåÿâí. ñõ.2aÍåÿâíàÿ ñõåìà:••• n+1•n!nfm22n+1n+1n+1nun+1m − um2 um+1 − 2um + um−1n−a= fmτh2- øàáëîí ñõåìû ⇒ íàäî ðåøàòü ñèñò. óð-íèé.≡ 0, ïîäñò. unm = λn eimα , λ(α), ïðîâåð., ÷òî |λ| ≤ 1 ∀ α.Ëåêöèÿ 22(9).∂u∂ 2u= a2 2 + f (x, t)∂t∂xu(x, 0) = u0 (x)u(0, t) = µ1 (t)u(X, t) = µ2 (t)t6TM, NXh=Mτ-0hXxτ=TNunm ≈ u(mh, nτ )ÁÁnun − 2unm + unm+1un+1óñë.m − um2 m−1n=a+ fmóñò.τh2ÁÁn+1n+1n+1nn+1u− 2um + um+1um − umáåçóñë.2 m−1n=a+ fmóñò.τh2a:=1ßâíàÿ ñõåìà89 n+1unm−1 − 2unm + unm+1um − unmn=+ fm2τh(∗)un0 = µn1 , n = 0, NunM = µn2 , n = 0, N u0 = u (mh) , m = 0, M0m?Õîðîøî áû u00 = u0 (0) = µ01 , åñëè íà÷.
ô-èè â 0 èì. îäèíàê. çíà÷åíèÿ, òî èìååì ¾ = ¿ ;ñ÷èòàåì, ÷òî ýòî èìååò ìåñòî áûòü, òîãäà â (∗) â í.ó. m = 1, Mh2Òåîðåìà: ñõåìà (∗) óñòîé÷èâà ïðè τ ≤2Äîê-âî: îáîçíà÷èì ρ = τ /h2 ,nn+1= (1 − 2ρ)unm + ρunm−1 + ρunm+1 + τ fmumn+1Ñ÷èòàåì max |un+1m | = um0 - âíóòð. òî÷êà (ò.å. max äîñòèãàåòñÿ íå íà ãðàíèöå) ⇒mun+1= (1 − 2ρ)unm ...mnnnkun+1 k = max |un+1=m | ≤ (1 − 2ρ)ku k + 2ρku k + τ kf k°mïî óñëîâèþ (1 − 2ρ) ≥ 0, (kun k = max |unm | - íîðìà íà n-íîì ñëîå)m°= kun k + τ kf n k⇒ kun+1 k ≤ kun k + τ kf n kåñëè max äîñòèãàåòñÿ íà ãðàíèöå ⇒max |un+1 | ≤ max(|µn+1|, |µn+1|)12mîáúåäèíÿÿ 2 ñëó÷àÿ:⇒ max |un+1 | ≤ max(|µn+1|, |µn+1|, kun k + τ kf n k)12mÍàì íàäî ä-òü: kuh kh ≤ C(kfh kh + kµ1h kh + kµ2h kh + ku0 kh )(k.kh − max çíà÷.)uh = yh + vh - ðåøåíèå óð-íèÿ èç (∗):yh : fh 6= 0 ðåøåíèå (∗) òàêîå, ÷òî íà÷.
è êðàåâ. óñëîâ =0: µ1 ≡ 0, µ2 ≡ 0, u0 ≡ 0;vh : fh = 0 ðåøåíèå (∗) òàêîå, ÷òî µ1 6= 0, µ2 6= 0, u0 6= 090kv n+1 k ≤ max(max |µk1 |, max |µk2 |, kv n k) ≤kkqku0 kky n+1 k ≤ ky n k + τ kf n k ≤≤ ky n−1 k + τ (kf n k + kf n−1 k) ≤ ...nP≤ ky 0 k + τkf k k ≤ T · kfh khk=1q0⇒ kuh kh ≤ max kun k ≤ max(max |µk1 |, max |µk2 |, ku0 k + T max kf k k)nkkkÅñëè τ > h2 /2 âîñïîëüçóåìñÿ ñïåêòð. ïðèçíàêîì óñòîé÷èâîñòè ⇒ ïîëó÷èì, ÷òî ñõåìàíåóñòîé÷èâà.¤.Íåÿâíàÿ ñõåìàn+1n+1nun+1un+1m−1 − 2um + um+1m − umn+1=+ fm2τhun0 = µn1 , n = 0, NunM = µn2 , n = 0, N 0um = u0 (mh) , m = 0, MÒåîðåìà: Ðåøåíèå óñòîé÷èâî.Ä-âî(∆)un+1m +τn+1n+1nn+1(−un+1m−1 + 2um − um+1 ) = um + τ fm , m = 1, M − 1h2Ðàññì-ì kun+1 k (ìàêñèì. çíà÷.)Ïóñòü un+1- ñ íàèìåíüøèì èíäåêñîì m, íà êîòîð.
äîñòèã-ñÿ íîðìà kun+1 km1. m = 0 (êðàåâ. óñëîâ.)m=M⇒ kun+1 k ≤ max(|µn+1|, |µn+1|,12)kun k + τ kf k{z}|äîáàâë. ïðîñòî òàê2. 1 < m < Mkun+1 k = |un+1m | (õîòèì îöåíèòü ÷/ç ìîäóëü ëåâ ÷àñòè (∆), à ïîòîì ïîëó÷èòü, ÷òîõîòèì:)91n+1|un+1m | > |um−1 |n+1|un+1m | ≥ |um+1 |n+1n+1n+1⇒ sgn(−un+1m−1 + 2um − um+1 ) = sgnumτn+1n+1(−un+1m−1 + 2um − um+1 )| =2hn+1= |unm + τ fm| ≤ kun k + τ kf n+1 k⇒ kun+1 k = |un+1m +1., 2. ⇒ kun+1 k ≤ max(|µn+1|, |µn+1|, kun k + τ kf k)12Àíàëîãè÷íî ÿâíîé ñõåìå ðàçáèâàåì ðåøåíèå íà 2 ÷àñòè è ò.ä.¤τn+1n+1nn+1(−un+1m−1 + 2um − um+1 ) = um + τ fm2h2≤m≤M −2:un+1m +m=1:ττ(2un+1− un+1) = un1 + τ f1n+1 + 2 · µn+11212hhàíàëîãè÷íîm=M −1:un+1+1M − 1 - íåèçâåñòíûõ un+1...un+11M −12ττ− 2 1 + h2h2τ −τ1+ 22hhτ0− 2h0τ− 2h2τ1+ 2h00τ− 2h......
0 ... 0 ... 0 Òðåõäèàãîíàëüíàÿ.Äèàãîíàëüíûé ýëåìåíò > ñóììû ìîäóëåé âíåäèàãîíàëüíûõ ýë-òîâ ⇒ íåâûðîæäåííàÿè ðåøàåòñÿ ì-äîì ïðîãîíêè.Çàòðàòû: 8M + O(M 2 )Ïîðÿäîê àïïðîêñèìàöèè O(τ + h2 )Ñõåìà ñ âåñàìèn+1n+1nunm−1 − 2unm + unm+1un+1un+1m−1 − 2um + um+1m − um=θ+ (1 − θ)+ ϕnm ; θ ∈ [0, 1]22τhhïîðÿäîê àïïðîêñèì: O(τ + h2 )Åñëè ðàññìîòðåòü àïïðîêñèìàöèþ â ò.
(m, n + 1/2) è ϕnm = f (mh, (n + 1/2)τ ), θ =1/2 ⇒ ïîðÿäîê O(τ 2 + h2 ) (Äëÿ ýòîé ñõåìû íåò îãðàíè÷åíèé íà øàãè - "õîðîøî")Îïðåäåëèì îïåðàòîð Λ:Λu|(x,t) =u(x + h, t) − 2u(x, t) + u(x − h, t)h292u - íåêîòîðàÿ ãëàäêàÿ ô-èÿ.Λunm =⇒unm+1 − 2unm + unm−1h2nun+1m − umnn= θΛun+1m + (1 − θ)Λum + ϕmτÎöåíèì ïîðÿäîê àïïðîêñèìàöèè â ò. (x, t + τ /2)¯¯τ 2 ∂ 3 u ¯¯u(x, t + τ ) − u(x, t)∂u ¯¯+=τ∂t ¯(x,t+τ /2) 24 ∂t3 ¯(x,t+ξ)¯τ ∂u ¯¯+ O(τ 2 ) =Λu|(x,t+τ /2±τ /2) = Λu|(x,t+τ /2) ± Λ¯2∂t(x,t+τ¯ /2)¯¯∂ 2 u ¯¯h2 ∂ 4 u ¯¯τ ∂u ¯¯=+± Λ+ O(τ 2 + h4 ).∂x2 ¯(x,t+τ /2) 12 ∂x4 ¯(x,t+τ /2) 2 ∂t ¯(x,t+τ /2)u(x, t + τ ) − u(x, t)− θΛu(x, t + τ ) − (1 − θ)Λu(x, t) − f (x, t + τ /2) =τÃ!¯¯∂u∂u ¯¯∂ 2 u ¯¯=− (θ + (1 − θ)) 2 ¯+− f (x, t + τ /2) +τ ( θ − 1/2 )Λ| {z }∂t∂x (x,t+τ /2)∂t ¯(x,t+τ /2)=0 ïðè θ=1/2{z}|=0 ò.ê. ðåø. çàäà÷è+O(τ 2 + h2 ) = O(τ 2 + h2 ) ïðè θ = 1/2Òåîðåìà: (á/ä) Åñëè θ ≥ 1/2, òî ýòà ñõåìà áåçóñëîâíî óñòîé÷èâà.Íå âõîäèò â ïðîãð. ýêç.Ïðèíöèï çàìîðîæåííûõ êîýôôèöèåíòîâ∂u∂ 2u= a(x, t) 2∂t∂xnun − 2unm + unm−1un+1m − um= a(mh, nτ ) m+1τh2Îáû÷íî a(x, t) > 0.Ôèêñèðóåì a â ò.
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