(Q51846055)

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explicit Runge–Kutta method

type of Runge-Kutta method

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{\begin{aligned}k_{1}&=f\left(t_{n},y_{n}\right)\\k_{2}&=f\left(t_{n}+c_{2}h,y_{n}+h\left(a_{21}k_{1}\right)\right)\\k_{3}&=f\left(t_{n}+c_{3}h,y_{n}+h\left(a_{31}k_{1}+a_{32}k_{2}\right)\right)\\&\vdots \\k_{s}&=f\left(t_{n}+c_{s}h,y_{n}+h\left(a_{s1}k_{1}+a_{s2}k_{2}+\cdots +a_{s,s-1}k_{s-1}\right)\right)\\y_{n+1}&=y_{n}+h\sum _{i=1}^{s}b_{i}k_{i}\end{aligned}}

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{\begin{array}{r|lllll}0&&&&&\\c_{2}&a_{21}&&&\\c_{3}&a_{31}&a_{32}&&\\\vdots &\vdots &&\ddots &\\c_{s}&a_{s1}&a_{s2}&\cdots &a_{s,s-1}\\\hline &b_{1}&b_{2}&\cdots &b_{s-1}&b_{s}\end{array}}

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