Runge-Kutta meetodid on arvutusmatemaatikas algoritmide pere harilike diferentsiaalvõrrandite (ja harilike diferentsiaalvõrrandite süsteemide ) ilmutatud või ilmutamata ligikaudse lahendi numbriliseks leidmiseks algtingimustega ülesande korral. Nad põhinevad iteratsioonil .
Meetodi algse kuju töötas välja saksa matemaatik Carl Runge 1895 ning seda üldistas Martin Wilhelm Kutta 1901 .
Pikemalt artiklis Klassikaline Runge-Kutta meetid 4. järku klassikaline Runge-Kutta meetod on nii laialt levinud, et seda nimetatakse sageli lihtsalt Runge-Kutta meetodiks.
Olgu meil Cauchy ülesanne
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{\displaystyle {\textbf {y}}'={\textbf {f}}(x,{\textbf {y}}),{\textbf {y}}(x_{0})={\textbf {y}}_{0}}
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{\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})}
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{\displaystyle {\textbf {k}}_{1}={\textbf {f}}\left(x_{n},{\textbf {y}}_{n}\right),}
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{\displaystyle {\textbf {k}}_{2}={\textbf {f}}\left(x_{n}+{h \over 2},{\textbf {y}}_{n}+{h \over 2}{\textbf {k}}_{1}\right),}
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{\displaystyle {\textbf {k}}_{3}={\textbf {f}}\left(x_{n}+{h \over 2},{\textbf {y}}_{n}+{h \over 2}{\textbf {k}}_{2}\right),}
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{\displaystyle {\textbf {k}}_{4}={\textbf {f}}\left(x_{n}+h,{\textbf {y}}_{n}+h{\textbf {k}}_{3}\right),}
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{\displaystyle {\textbf {}}h}
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{\displaystyle {\textbf {}}x}
See meetod on 4. järku, st viga igal sammul on
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{\displaystyle O(h^{5})}
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{\displaystyle O(h^{4})}
Otseste Runge-Kutta meetodite pere on 4. järku Runge-Kutta meetodi üldistus. See on antud valemitega
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{\displaystyle {\textbf {y}}_{n+1}={\textbf {y}}_{n}+h\sum _{i=1}^{s}b_{i}{\textbf {k}}_{i},}
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{\displaystyle {\textbf {k}}_{1}=f(x_{n},{\textbf {y}}_{n}),\,}
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{\displaystyle {\textbf {k}}_{2}=f(x_{n}+c_{2}h,{\textbf {y}}_{n}+a_{21}h{\textbf {k}}_{1}),\,}
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{\displaystyle {\textbf {k}}_{3}=f(x_{n}+c_{3}h,{\textbf {y}}_{n}+a_{31}h{\textbf {k}}_{1}+a_{32}h{\textbf {k}}_{2}),\,}
⋮
{\displaystyle \vdots }
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{\displaystyle {\textbf {k}}_{s}=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}).}
Konkreetse meetodi määravad arv
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{\displaystyle s}
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{\displaystyle b_{i},a_{ij}}
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{\displaystyle c_{2}}
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{\displaystyle a_{21}}
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{\displaystyle c_{3}}
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{\displaystyle a_{31}}
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{\displaystyle a_{32}}
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{\displaystyle \vdots }
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{\displaystyle \vdots }
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{\displaystyle \ddots }
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{\displaystyle c_{s}}
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{\displaystyle a_{s1}}
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{\displaystyle a_{s2}}
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{\displaystyle \cdots }
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{\displaystyle a_{s,s-1}}
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{\displaystyle b_{1}}
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{\displaystyle b_{2}}
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{\displaystyle \cdots }
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{\displaystyle b_{s-1}}
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{\displaystyle b_{s}}
Runge-Kutta meetodi koefitsiendid peavad rahuldama tingimusi
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{\displaystyle \sum _{j=1}^{i-1}a_{ij}=c_{i}}
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{\displaystyle i=2,\ldots ,s}
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{\displaystyle p}
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{\displaystyle {\bar {\textbf {y}}}(h+x_{0})-{\textbf {y}}(h+x_{0})=O(h^{p+1})}
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{\displaystyle {\bar {\textbf {y}}}(h+x_{0})}
Pikemalt artiklis Klassikaline Runge-Kutta meetid
