Н.И. Ионкин - Электронные лекции (2009) (1135239), страница 12
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Ïðèâå-äåì ïðèìåðû ðàçíîñòíûõ ìåòîäîâ Ðóíãå-Êóòòà, èìåþùèõ òðåòèé è ÷åòâåðòûé ïîðÿäîê ïîãðåøíîñòè àïïðîêñèìàöèè.Ïðèìåð.Ñõåìà Ðóíãå-Êóòòà ÷åòâåðòîãî ïîðÿäêà.1yn+1 − yn= (K1 + 2K2 + 2K3 + K4 )τ6K1 = f (tn , yn )K2 = f (tn + 0.5τ, yn + 0.5τ K1 )K3 = f (tn + 0.5τ, yn + 0.5τ K2 )K4 = f (tn + τ, yn + τ K3 )Äàííàÿ ñõåìà èìååò ÷åòâåðòûé ïîðÿäîê àïïðîêñèìàöèè ïîÏðèìåð.τ : ψn = O(τ 4 ).Ñõåìà Ðóíãå-Êóòòà òðåòüåãî ïîðÿäêà.yn+1 − yn1= (K1 + 4K2 + K3 )τ6K1 = f (tn , yn )K2 = f (tn + 0.5τ, yn + 0.5τ K1 )K3 = f (tn + τ, yn − τ K1 − 2τ K2 )Äàííàÿ ñõåìà èìååò òðåòèé ïîðÿäîê àïïðîêñèìàöèè ïî2τ : ψn = O(τ 3 ).Îöåíêà òî÷íîñòè íà ïðèìåðå 2-õ ýòàïíîãîìåòîäà Ðóíãå-Êóòòà(dudt= f (t, u(t)),u(0) = u0t>0yn+1 − yn= (1 − σ)f (tn , yn ) + σf (tn + at, yn + aτ f (tn , yn ))τ(1)Îöåíêà òî÷íîñòè íà ïðèìåðå 2-õ ýòàïíîãî ìåòîäà Ðóíãå-Êóòòà111y0 = u0tn ∈ ωτσ - ïàðàìåòð, â êà÷åñòâå êîòîðîãî ìîæíî âûáèðàòü ëþáîå ÷èñëî, ëèøüáû âûïîëíÿëîñü óñëîâèå âòîðîé ïîãðåøíîñòè àïïðîêñèìàöèè.
Îáû÷íîâûáèðàþòσ ∈ [0, 1].à - íåêîòîðàÿ êîíñòàíòà. Áóäåì ðàññìàòðèâàòüa ≥ 0,íî, âîîáùå ãî-âîðÿ, ýòî íåîáÿçàòåëüíî.Ââåäåì ôóíêöèþ ïîãðåøíîñòèzn :zn = yn − u(tn ) = yn − un ⇒(2)un+1 − unzn+1 − zn=−+(1−σ)f (tn , yn )+σf (tn +aτ, yn +atf (tn , yn ))ττ(3)Äëÿ ñõîäèìîñòè íóæíî ïîêàçàòü, ÷òî:|zn | → 0,Ïîêàæåì, ÷òî|zn | ≤ M τ 2 ,n→∞ãäå M íå çàâèñèò îòτzn+1 − znun+1 − un=−+ (1 − σ)f (tn , un )+ττσf (tn + aτ, un + aτ f (tn , un )) − (1 − σ)f (tn , un )+(1 − σ)f (tn , yn ) − σf (tn + aτ, un + aτ f (tn , un ))+(2)σf (tn + aτ, yn + aτ f (tn , yn )) = ψn + φ(1)n + φnãäå(1)(2)ψn , φn , φnψn = −îáîçíà÷åíû ñëàãàåìûå:un+1 − un+ (1 − σ)f (tn , un ) + σf (tn + aτ, un + aτ f (tn , un )),τφ(1)n = (1 − σ)(f (tn , yn ) − f (tn , un )),φ(2)nhi= σ f (tn + aτ, yn + aτ f (tn , yn )) − f (tn + aτ, un + aτ f (tn , un )) .(4)Îöåíêà òî÷íîñòè íà ïðèìåðå 2-õ ýòàïíîãî ìåòîäà Ðóíãå-Êóòòà112Ââåäåì äîïóùåíèå: ôóíêöèÿ f ïî âòîðîìó àðãóìåíòó óäîâëåòâîðÿåòóñëîâèþ Ëèïøèöà ñ êîíñòàíòîé L.
Îöåíèì, èñõîäÿ èç ýòîãî äîïóùåíèÿ,(1)(2)φn è φn :|φ(1)n | ≤ (1 − σ)|f (tn , yn ) − f (tn , un )| ≤ (1 − σ)L|yn − un | = (1 − σ)L|zn |,|φ(1)n | ≤ σL|yn + aτ f (tn , yn ) − un + aτ f (tn , un )| ≤≤ σL(|yn − un | + |{z}a τ L|yn − un |) = σL(1 + aτ L)| yn − un || {z }≥0Èç (3)zn⇒(2)zn+1 = zn + τ ψn + τ φ(1)n + τ φn|zn+1 | ≤ |zn | + τ |ψn | + τ (1 − σ)L|zn | + σL|zn | + σaτ L2 |zn | =τ |ψn | + (1 + τ L + τ 2 aσL2 )|zn |σa ≤ 0, 5, çàìåòèâ, ÷òî 1+τ L+0, 5τ 2 L2 ÿâëÿþòñÿ ïåðâûìèτLðàçëîæåíèÿ ïî Òåéëîðó ôóíêöèè e :Ðàññìîòðèì÷ëåíàìè|zn+1 | ≤ τ |ψn | + (1 + τ L + 0, 5τ 2 L2 )|zn | ≤ eτ L |zn | + τ |ψn |Îáîçíà÷èìeτ L = ρ.(5)Ïîëó÷èì îöåíêó:|zn+1 | ≤ ρ|zn | + τ |ψn |(6)Ñîîòíîøåíèå (6) ìîæíî ðàññìîòðåòü êàê ðåêóððåíòíóþ ôîðìóëó. Ëåãêî âèäåòü, ÷òî:zn+1 ≤ ρn+1 |z0 | +nXρn−j τ |ψj |j=0|zn+1 | ≤ max |ψj |0≤j≤nnXρn−j τ ≤ tn+1 eLtn+1 max |ψj |j=00≤j≤nÎêîí÷àòåëüíî, ïîëó÷àåì:|zn+1 | ≤ M max |ψj |,0≤j≤nM íå çàâèñèò îòτ(7)Âèäíî, ÷òî òî÷íîñòü áóäåò ñîâïàäàòü ñ ïîðÿäêîì ïîãðåøíîñòè àïïðîêñèìàöèè, à èìåííî:Ìíîãîøàãîâûå ðàçíîñòíûå ìåòîäû1.113σa = 0, 5 ⇒ ψ = O(τ 2 ) ⇒ |zn | = O(τ 2 ),ò.å.
èìååì âòîðîé ïîðÿäîêïîãðåøíîñòè.2.σ = 0, ∀a ⇒ ψ = O(τ ),|zn | ≤ M1 τ,M1íå çàâèñèò îòτ,ïîëó÷à-åì ïåðâûé ïîðÿäîê òî÷íîñòè.3Ìíîãîøàãîâûå ðàçíîñòíûå ìåòîäû(dudt= f (t, u(t)),u(0) = u0t>0(1)ωτ = tn = nτ, τ > 0, n = 0, 1, . . . .yk = y(tk ), fk = f (t, yk ).Ââåäåì ñåòêóÎáîçíà÷èìÎïðåäåëåíèå. Ëèíåéíûì m-øàãîâûì ðàçíîñòíûì ìåòîäîì ðåøåíèÿçàäà÷è () íàçûâàåòñÿ ìåòîä, çàïèñàííûé óðàâíåíèåì:mXakk=0ãäåak , bk- ÷èñëà,τ > 0.τyn−k =mXbk fn−k ,(2)k=0Ïðè ýòîìa0 6= 0, bm 6= 0, n = m, m + 1, .
. .b0 = 0, òî (2) - ÿâíûé ìåòîä. Åñëè b0 6= 0, òî (2) - íåÿâíûé ìåòîä.Äëÿ íà÷àëà âû÷èñëåíèé ïî ôîðìóëå (2) íåîáõîäèìû çíà÷åíèÿ y0 , . . . , ym−1- ò.í. Ðàçãîííûé ýòàï. Òàê êàê ôîðìóëà (2) îäíîðîäíà ïî ak è bk , òî ïîPmëàãàþòk=0 = 1(óñëîâèå íîðìèðîâêè).Íåÿâíûé m-øàãîâûé ðàçíîñòíûé ìåòîä çàïèñûâàåòñÿ â âèäåÅñëèa0yn − b0 f (tn , yk ) = F (yn−1 , yn−2 , . . . , yn−m )τmmXXakF =bk fn−k −yn−kτk=1k=1(3)Óðàâíåíèå (3) ðåøàåòñÿ ÷àùå âñåãî ìåòîäîì Íüþòîíà, ïðè÷åì â êà÷å(0)ñòâå yn áåðåòñÿ yn−1 .  ÿâíîì ðàçíîñòíîì ìåòîäå çíà÷åíèÿ yn íàõîäÿòñÿïî ÿâíîé ôîðìóëåyn =mmXτ Xakbk fn−k −yn−ka0 k=1τk=1Ìíîãîøàãîâûå ðàçíîñòíûå ìåòîäû114Îöåíèì ïîãðåøíîñòü àïïðîêñèìàöèè íà ðåøåíèèψn = −mXakτk=0un−k +mXun = u(tn − kτ ) =pX(−kτ )ll!l=0f (tn−k , un−k ) =u0n−kpmXak X (−kτ )lτk=0=l=0l!=l!u(l) (tn ) +mXbku(l+1) (tn ) + O(τ p )p−1Xñäâèã èíäåêñîâo=−pmXXak (−kτ )ll=0 k=0pmXXbkl=1 k=0−mXk=0u(l+1) (tn ) = O(τ p ) =l=0k=0n(4)u(l) (tn ) + O(τ p+1 )p−1X(−kτ )ll=0ψn = −bk f (tn−k , un−k )k=0τl!u(l)n +(−kτ )l−1 (l)u + O(τ p ) =(l − 1)! npm hX(l)Xun iakl−1(−kτ ) (ak k + lbk )un ++ O(τ p )τl(l−1)!k=0 l=1Óñëîâèå àïïðîêñèìàöèè:mXak = 0k=0Äëÿ äîñòèæåíèÿ àïïðîêñèìàöèè ïîðÿäêà p äîëæíî áûòü âûïîëíåíîñîîòíîøåíèå:mXk l−1 (ak k + lbk ) = 0,l = 1, 2, .
. . , pk=0 ìíîãîøàãîâîì ìåòîäå 2m+2 íåèçâåñòíûõ -a0 , a1 , . . . , am , b0 , . . . , bm ,è p+2 óðàâíåíèé. ×òîáû ñèñòåìà íå áûëà ïåðåîïðåäåëåííîé, äîëæíîâûïîëíÿòüñÿp ≤ 2m ⇒íàèâûñøèé ïîðÿäîê àïïðîêñèìàöèè ðàâåí 2m.Ïîíÿòèå óñòîé÷èâîñòè ðàçíîñòíûõ ìåòîäîâ115Òàêèì îáðàçîì, äëÿ äîñòèæåíèÿ ïîðÿäêà ïîãðåøíîñòè àïïðîêñèìàöèèpäîëæíû âûïîëíÿòüñÿ ñëåäóþùèå ñîîòíîøåíèÿ:a0 = −mXakk=1b0 = 1 −mXbkk=1mXk l−1 (ak k + lbk ) = 0,l = 1, 2, . .
. , pk=04Ïîíÿòèå óñòîé÷èâîñòè ðàçíîñòíûõ ìåòîäîâÐàññìîòðèì çàäà÷ó Êîøè:(dudt= f (t, u(t)),u(0) = u0 .t > 0,(1)Ðàññìîòðèì äëÿ ïðèìåðà òàêóþ ñõåìó:yn = qyn−1 ,q ∈ C,n = 0, 1, . . . ;Ïðèäàäèìynâîçìóùåíèåy0q = const,çàäàí.δn :ỹn = yn + δn .ÒîãäàÅñëèỹn+1 = q ỹn = qyn + qδn = yn+1 + δn+1 , ãäå δn+1 = qδn .|q| > 1, òî δn íàðàñòàåò, ñëåäîâàòåëüíî, îá óñòîé÷èâîñòèãîâî-ðèòü íåëüçÿ.Ðàññìîòðèì ìîäåëüíóþ çàäà÷ó:(dudt+ λu(t) = 0,u(0) = u0 .t > 0,(2)Ïîíÿòèå óñòîé÷èâîñòè ðàçíîñòíûõ ìåòîäîâÅå ðåøåíèå èìååò âèäu(t) = u0 e−λt .Åñëèλ > 0,116òî |u(t)|≤ |u0 |,ò.å.èìååò ìåñòî óñòîé÷èâîñòü ïî íà÷àëüíîìó óñëîâèþ.Óñòîé÷èâîñòü âíóòðåííåå ñâîéñòâî ðàçíîñòíîé ñõåìû.
Ðàçíîñòíàÿñõåìà íå îáÿçàòåëüíî ñîõðàíÿåò óñòîé÷èâîñòü èñõîäíîé çàäà÷è.Ðàññìîòðèì ÿâíóþ ñõåìó Ýéëåðà:(yn+1 −ynτ= f (tn , yn ),y0 = u0 .(3)Çàïèøåì åå äëÿ ìîäåëüíîé çàäà÷è:yn+1 − yn+ λyn = 0.τÂûðàçèìyn+1 :yn+1 = yn − τ λyn = (1 − τ λ)yn .Îáîçíà÷èìq = 1 − τ λ.Òîãäàyn+1 = qyn . Òàêèì îáðàçîì,|q| ≤ 1, ò.å.äëÿ óñòîé÷è-âîñòü íåîáõîäèìî, ÷òîáû âûïîëíÿëîñü1 − τ λ ≥ −1,0 < τ λ ≤ 2.Òàêèì îáðàçîì, äëÿ òîãî, ÷òîáû ÿâíàÿ ñõåìà Ýéëåðà áûëà óñòîé÷èâîé(äëÿ ìîäåëüíîé çàäà÷è), íåîáõîäèìî âûïîëíåíèå óñëîâèÿ0<τ ≤2.λ(4)Ýòî îçíà÷àåò, ÷òî ÿâíàÿ ñõåìà Ýéëåðà ÿâëÿåòñÿ óñëîâíî óñòîé÷èâîé(äëÿ ìîäåëüíîé çàäà÷è).Ðàññìîòðèì íåÿâíóþ ñõåìó Ýéëåðà:yn+1 − yn= f (tn+1 , yn+1 ).τÏåðåïèøåì åå:yn+1 + τ f (tn+1 , yn+1 ) = yn .(5)Äëÿ ðåøåíèÿ íåëèíåéíîãî óðàâíåíèÿ (5) îáû÷íî ïðèìåíÿåòñÿ ìåòîäÍüþòîíà, â êà÷åñòâå íà÷àëüíîãî ïðèáëèæåíèÿ äëÿ íàõîæäåíèÿïîëüçóþòyn .yn+1èñ-Ïîíÿòèå óñòîé÷èâîñòè ðàçíîñòíûõ ìåòîäîâ117Ïåðåïèøåì (5) äëÿ ìîäåëüíîé çàäà÷è:yn+1 + τ λyn+1 = yn ,(1 + τ λ)yn = tn+1 ,yn+1 = qyn ,Çàìåòèì, ÷òî|q| < 1ïðèq=τ > 0, λ > 0.1.1 + τλÝòî çíà÷èò, ÷òî íåÿâíàÿ ñõåìàÝéëåðà ÿâëÿåòñÿ àáñîëþòíî óñòîé÷èâîé (äëÿ ìîäåëüíîé çàäà÷è).Òàêèì îáðàçîì, äëÿ óñòîé÷èâîé äèôôåðåíöèàëüíîé çàäà÷è ñóùåñòâóþò êàê óñòîé÷èâûå, òàê è íåóñòîé÷èâûå ñõåìû.Ðàññìîòðèì ïðîèçâîëüíûémXakk=0τyn−k =mXm-øàãîâûébk fn−k ,ðàçíîñòíûé ìåòîä:y0 , .
. . , ym−1çàäàíû.(6)k=0Çàïèøåì åãî äëÿ ìîäåëüíîé çàäà÷è:mXakk=0yn−k + λτmXmXbk yn−k = 0,(7)k=0(ak + τ λbk )yn−k = 0.k=0Áóäåì èñêàòü ðåøåíèå ýòîãî óðàâíåíèÿ â âèäåyj = q j .Ïîäñòàâèì ýòî (7):mX(ak + τ λbk )q n−k = 0.k=0Ðàçäåëèì îáå ÷àñòè ýòîãî óðàâíåíèÿ íàFm (τ, q) =q m−n ,ïîëó÷èìmX(ak + τ λbk )q m−k = 0.(8)k=0Óðàâíåíèå (8) íàçûâàåòñÿ õàðàêòåðèñòè÷åñêèì óðàâíåíèåì.
Äëÿ óñòîé÷èâîñòè íåîáõîäèìî, ÷òîáû åãî êîðíè ïî ìîäóëþ íå ïðåâîñõîäèëè 1 (èíà÷å ðåøåíèå áóäåò íåîãðàíè÷åííî íàðàñòàòü). Îäíàêî, íàõîæäåíèå êîðíåéÏîíÿòèå óñòîé÷èâîñòè ðàçíîñòíûõ ìåòîäîâ118óðàâíåíèÿ (8) òðóäíàÿ çàäà÷à, è îáû÷íî ðàññìàòðèâàþò áîëåå ïðîñòîåóðàâíåíèå:Fm (0, q) =mXak q m−k = 0.(9)k=0Óðàâíåíèå (9), òàêæå êàê è óðàâíåíèå (8), íàçûâàþò õàðàêòåðèñòè÷åñêèì óðàâíåíèåì.Îïðåäåëåíèå. Ãîâîðÿò, ÷òî ðàçíîñòíàÿ ñõåìàâèþ(α),(6) óäîâëåòâîðÿåò óñëî-åñëè âñå êîðíè õàðàêòåðèñòè÷åñêîãî óðàâíåíèÿ (9) ëåæàòâíóòðè èëè íà ãðàíèöå åäèíè÷íîãî êðóãà êîìïëåêñíîé ïëîñêîñòè, ïðè÷åì íà ãðàíèöå íåò êðàòíûõ êîðíåé.Òåîðåìà. Ïóñòü ðàçíîñòíàÿ ñõåìà|fn0 | ≤ Lïðè0 ≤ tn = τ n ≤ T .(α)èÒîãäà äëÿ ëþáîãî äîñòàòî÷íî ìàëîãîτ(6) óäîâëåòâîðÿåò óñëîâèþñïðàâåäëèâî|yn − un | ≤ MnX!τ |ψj | + max |yi − u(ti )| ,0≤i≤m−1j=mãäåMíå çàâèñèò îòτ, ψj(10) ïîãðåøíîñòü àïïðîêñèìàöèè ðàçíîñòíîãîìåòîäà (6) íà ðåøåíèå çàäà÷è (??).Çàìå÷àíèå.
Ìåòîä Àäàìñà óäîâëåòâîðÿåò óñëîâèþ (α):a0 = −a1 = 1,myn − yn−1 X=bk fn−k .τk−0Õàðàêòåðèñòè÷åñêîå óðàâíåíèå èìååò âèä:q n − q n−1 = 0,îíî èìååò êîðíèq=0èq = 1,ïðè÷åìq=1 íåêðàòíûé êîðåíü.Çàìå÷àíèå. Äëÿ íåÿâíûõ ñõåì íàèâûñøèé ïîðÿäîê ïîãðåøíîñòè àïïðîêñèìàöèèp ≤ 2m.Äëÿ ÿâíûõ ñõåìp ≤ 2m − 1.Îäíàêî, ñõåìû âûñîêîãî ïîðÿäêà íå óäîâëåòâîðÿþò óñëîâèþ(α),ò.å. íå ÿâëÿþòñÿ óñòîé÷èâûìè.
Íàèâûñøèé ïîðÿäîê ïîãðåøíîñòè àïïðîêñèìàöèè äëÿ ñõåì, óäîâëåòâîðÿþùèõ óñëîâèþ(α),ñëåäóþùèé:Ïîíÿòèå óñòîé÷èâîñòè ðàçíîñòíûõ ìåòîäîâ1191. Äëÿ íåÿâíûõ ñõåì:(a) Åñëè m ÷åòíî, òîp ≤ m + 2.(b) Åñëè m íå÷åòíî, òîp ≤ m + 1.p ≤ m.2. Äëÿ ÿâíûõ ñõåìÇàìå÷àíèå. Ãîâîðèòü îá óñëîâíîé èëè áåçóñëîâíîé óñòîé÷èâîñòè íåèìååò ñìûñëà. Îíà âñåãäà óñëîâíàÿ, ò.ê. ðàññìàòðèâàþòñÿ ìàëûåτ.Çàäà÷à. Äîêàçàòü, ÷òî äëÿ ñõåìûyn + 4yn−1 − 5yn−22fn−1 + fn−2=6τ3èìååò ìåñòîψn = O(τ 3 ).Ðåøåíèå.ψn = −un + 4un−1 − 5un−2 2fn−1 + fn−2+.6τ3Çàïèøåì óñëîâèÿ, íàëàãàåìûå íà ìíîãîøàãîâûé ðàçíîñòíûé ìåòîääëÿ òîãî, ÷òîáû ïîãðåøíîñòü àïïðîêñèìàöèè èìåëà ïîðÿäîê 3:mPbk ,b=1−0k=1mPak ,a0 = −mk=1Pak k = −1,k=1mP k l−1 (ak k + bk ) = 0, l = 2, 3.k=0 íàøåì ñëó÷àå,m = 2, a0 = 16 , a1 = 23 , a2 = − 65 , b0 = 0, b1 = 23 , b2 =1.
Âûïèñàííûå óñëîâèÿ, êàê ëåãêî ïðîâåðèòü, âûïîëíÿþòñÿ. Òàêèì îá33ðàçîì, ψn = O(τ ).Ðàññìîòðåííàÿ â ïðåäûäóùåé çàäà÷å ñõåìà íåóñòîé÷èâà. Äåéñòâèòåëüíî, äëÿ íåå õàðàêòåðèñòè÷åñêîå óðàâíåíèå èìååò âèä:q 2 + 4q − 5 = 0.Ýòî óðàâíåíèå èìååò êîðíèq1 = 1, q2 = −5.ðàçíîñòíûé ìåòîä íå óäîâëåòâîðÿåò óñëîâèþÒ.ê.(α).|q2 | > 1,òî äàííûéÆåñòêèå ñèñòåìû ÎÄÓ5120Æåñòêèå ñèñòåìû ÎÄÓÐàññìîòðèì ñèñòåìó ÎÄÓdu1 dt + a1 u1 (t) = 0, t > 0,du2+ a2 u2 (t) = 0, t > 0,dtu1 (0) = u10 , u2 (0) = u20 , a1 > 0, a2 > 0.(1)Ðåøåíèå èìååò âèä:u1 (t) = u10 e−a1 t ,u2 (t) = u20 e−a2 t .Ïóñòüa1 >> a2 .íåêîòîðîãî ìîìåíòàÒîãäà òàêàÿ ñèñòåìà ÎÄÓ íàçûâàåòñÿ æåñòêîé. Ñt∗ðåøåíèåu2 (t) ìàëî îòëè÷àåòñÿ îò 0.
Îäíàêî, åñëèìû ðåøàåì ýòó ñèñòåìó ïðè ïîìîùè ÿâíîé ñõåìû Ýéëåðà, òî íàì íóæíî22} = a22 . Ýòî áóäåò âåñüìà ìàëåíüêèé øàã,èñïîëüçîâàòü øàã τ ≤ min{ ,a1 a2∗èáî a1 >> a2 . Íî ñ íåêîòîðîãî ìîìåíòà t u2 ìîæíî íå ñ÷èòàòü, ò.å.èñïîëüçîâàíèå ìàëåíüêîãî øàãà èçëèøíå. Òàêèì îáðàçîì, ÿâíûå ñõåìûäëÿ æåñòêèõ ñèñòåì ÎÄÓ íå ãîäÿòñÿ. Åñëè èñïîëüçîâàòü íåÿâíóþ ñõåìó,òî ìîæíî âçÿòü áîëåå êðóïíûé øàã.A(m · m)ñ ïîñòîÿííûìèdu+ Au(t) = 0, t > 0dt(t)÷èñëàìè, u= (u1 (t), u2 (t), . .
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