Metodi Runge Kutti vazhliva grupa chiselnih metodiv rozv yazuvannya sistem zvichajnih diferencialnih rivnyan Nazvani na chest nimeckih matematikiv Karla Runge i Martina Kutti yaki vidkrili ci metodi Zmist 1 Klasichnij metod Runge Kutti 4 go poryadku 2 Pryami metodi Runge Kutti 3 Priklad rozv yazannya v seredovishi MATLAB 4 Div takozh 5 LiteraturaKlasichnij metod Runge Kutti 4 go poryadku RedaguvatiMetod Runge Kutti 4 go poryadku nastilki shiroko rozpovsyudzhenij sho jogo chasto nazivayut prosto metodom Runge Kutti abo RK4 Rozglyanemo zadachu Koshi dlya sistemi diferencialnih rivnyan dovilnogo poryadku sho zapisuyetsya u vektornij formi yak y f x y y x 0 y 0 displaystyle textbf y textbf f x textbf y textbf y x 0 textbf y 0 nbsp dd Todi znachennya nevidomoyi funkciyi v tochci x n 1 displaystyle x n 1 nbsp obchislyuyetsya vidnosno znachennya v poperednij tochci x n displaystyle x n nbsp za formuloyu y n 1 y n h 6 k 1 2 k 2 2 k 3 k 4 displaystyle textbf y n 1 textbf y n h over 6 textbf k 1 2 textbf k 2 2 textbf k 3 textbf k 4 nbsp dd x n 1 x n h displaystyle x n 1 x n h nbsp dd de h displaystyle h nbsp krok integruvannya a koeficiyenti k n displaystyle textbf k n nbsp rozrahovuyutsya takim chinom k 1 f x n y n displaystyle textbf k 1 textbf f left x n textbf y n right nbsp k 2 f x n h 2 y n h 2 k 1 displaystyle textbf k 2 textbf f left x n h over 2 textbf y n h over 2 textbf k 1 right nbsp k 3 f x n h 2 y n h 2 k 2 displaystyle textbf k 3 textbf f left x n h over 2 textbf y n h over 2 textbf k 2 right nbsp k 4 f x n h y n h k 3 displaystyle textbf k 4 textbf f left x n h textbf y n h textbf k 3 right nbsp Ce metod 4 go poryadku tobto pohibka na kozhnomu kroci stanovit O h 5 displaystyle O h 5 nbsp a sumarna pohibka na kincevomu intervali integruvannya ye velichinoyu O h 4 displaystyle O h 4 nbsp Pryami metodi Runge Kutti RedaguvatiGrupa pryamih metodiv Runge Kutti ye uzagalnennyam metodu Runge Kutti 4 go poryadku Nablizhennya zadayetsya formulami y n 1 y n h i 1 s b i k i displaystyle textbf y n 1 textbf y n h sum i 1 s b i textbf k i nbsp de k 1 f x n y n displaystyle textbf k 1 textbf f x n textbf y n nbsp k 2 f x n c 2 h y n a 21 h k 1 displaystyle textbf k 2 textbf f x n c 2 h textbf y n a 21 h textbf k 1 nbsp k 3 f x n c 3 h y n a 31 h k 1 a 32 h k 2 displaystyle textbf k 3 textbf f x n c 3 h textbf y n a 31 h textbf k 1 a 32 h textbf k 2 nbsp displaystyle vdots nbsp dd dd k s f x n c s h y n a s 1 h k 1 a s 2 h k 2 a s s 1 h k s 1 displaystyle textbf k s textbf f x n c s h textbf y n a s1 h textbf k 1 a s2 h textbf k 2 cdots a s s 1 h textbf k s 1 nbsp Konkretnij metod viznachayetsya chislom s displaystyle s nbsp i koeficiyentami b i a i j displaystyle b i a ij nbsp i c i displaystyle c i nbsp Ci koeficiyenti chasto vporyadkovuyut v tablicyu 0c 2 displaystyle c 2 nbsp a 21 displaystyle a 21 nbsp c 3 displaystyle c 3 nbsp a 31 displaystyle a 31 nbsp a 32 displaystyle a 32 nbsp displaystyle vdots nbsp displaystyle vdots nbsp displaystyle ddots nbsp c s displaystyle c s nbsp a s 1 displaystyle a s1 nbsp a s 2 displaystyle a s2 nbsp displaystyle cdots nbsp a s s 1 displaystyle a s s 1 nbsp b 1 displaystyle b 1 nbsp b 2 displaystyle b 2 nbsp displaystyle cdots nbsp b s 1 displaystyle b s 1 nbsp b s displaystyle b s nbsp Dlya koeficiyentiv metodu Runge Kutti mayut spravdzhuvatis umovi j 1 i 1 a i j c i displaystyle sum j 1 i 1 a ij c i nbsp dlya i 2 s displaystyle i overline 2 s nbsp Yaksho mi hochemo shob metod mav poryadok p displaystyle p nbsp to varto tak samo zabezpechiti umovu y h x 0 y h x 0 O h p 1 displaystyle bar textbf y h x 0 textbf y h x 0 O h p 1 nbsp de y h x 0 displaystyle bar textbf y h x 0 nbsp nablizhennya otrimane za metodom Runge Kutti Pislya bagatorazovogo diferenciyuvannya cya umova peretvoritsya v sistemu polinomialnih rivnyan rozv yazki yakoyi ye koeficiyentami metodu Pryami metodi rozv yazku zhorstkih diferencialnih rivnyan ta yih sistem neefektivni vnaslidok rizkogo zbilshennya kilkosti krokiv obchislen pri zmenshenni kroku integruvannya h displaystyle h nbsp chi zrostannya pohibki pri nedostatno malomu kroci h displaystyle h nbsp Nehaj pohibka maye poryadok k displaystyle k nbsp Nablizhene znachennya y x displaystyle y x nbsp obchislene u tochci x displaystyle x nbsp iz velichinoyu kroku h displaystyle h nbsp poznachayetsya Y x h displaystyle Y x h nbsp Todi u tochci x x 0 2 n h displaystyle x x 0 2nh nbsp y x Y x h A 2 n h k 1 A x x 0 h k y x Y x 2 h A n 2 h k 1 A x x 0 2 k h k displaystyle y x Y x h approx A2nh k 1 A x x 0 h k quad quad y x Y x 2h approx An 2h k 1 A x x 0 2 k h k nbsp tobto Y x 2 h Y x h A x x 0 1 2 k h displaystyle Y x 2h Y x h approx A x x 0 1 2 k h nbsp ta vidpovidno y x Y x h Y x h Y x 2 h 2 k 1 displaystyle y x Y x h approx frac Y x h Y x 2h 2 k 1 nbsp Pomilka pri kroci h displaystyle h nbsp virazhayetsya cherez nablizheni znachennya pri krokah h displaystyle h nbsp ta 2 h displaystyle 2h nbsp Bagatokrokovi metodi vikoristovuyut dlya obchislennya nastupnogo znachennya y j 1 displaystyle y j 1 nbsp lishe informaciyu z napivintervalu y j y j 1 displaystyle y j y j 1 nbsp Bagatokrokovi metodi bazuyutsya na zamini diferencialnogo rivnyannyay x f x y x 0 y a s displaystyle y x f x y x 0 quad quad y a s nbsp za stalogo kroku h displaystyle h nbsp riznicevim rivnyannyam poryadku k displaystyle k nbsp j 0 k a j k y h j 0 k b j k f x k j y k j 0 a k k 0 a 0 k 2 b 0 k 2 0 n 0 1 displaystyle sum j 0 k a j k y h sum j 0 k b j k f x k j y k j 0 quad quad a k k neq 0 quad quad a 0 k 2 b 0 k 2 neq 0 quad quad n 0 1 nbsp y 0 s y 1 y k 1 displaystyle y 0 s y 1 y k 1 nbsp zadani znachennya Kozhnij sposib takogo tipu viznachayetsya polinomamip k z a k k z k a k 1 k z k 1 z 0 k s k z b k k z k b k 1 k z k 1 b 0 k displaystyle p k z a k k z k a k 1 k z k 1 z 0 k quad quad sigma k z b k k z k b k 1 k z k 1 b 0 k nbsp Yaksho stepin s k displaystyle sigma k nbsp menshe stepeni p k displaystyle p k nbsp to govoryat pro yavnij abo vidchinenij metod yaksho stepeni rivni to pro neyavnij zachinenij Priklad rozv yazannya v seredovishi MATLAB RedaguvatiRozv yazannya sistem diferencialnih rivnyan metodom Runge Kutti ye odnim z najbilsh poshirenih chislovih metodiv rozv yazannya v tehnici V seredovishi MATLAB Octave dosit poshirena i zruchna mova programuvannya dlya tehnichnih obchislen realizovanij odin z jogo riznovidiv metod Dormanda Prinsa Div takozh RedaguvatiMetod Ejlera Metod Adamsa Metod Henrichi Metod MilnaLiteratura RedaguvatiWilliam H Press Brian P Flannery Saul A Teukolsky William T Vetterling Numerical Recipes in C Cambridge UK Cambridge University Press 1988 Rozdili 16 1 i 16 2 Otrimano z https uk wikipedia org w index php title Metod Runge Kutti amp oldid 37712643
