(Q51846061)

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implicit Runge–Kutta method

type of Runge-Kutta method

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{\begin{aligned}k_{i}&=f\left(t_{n}+c_{i}h,y_{n}+h\sum _{j=1}^{s}a_{ij}k_{j}\right),\quad i=1,\ldots ,s\\y_{n+1}&=y_{n}+h\sum _{i=1}^{s}b_{i}k_{i}\end{aligned}}

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{\begin{array}{c|cccc}c_{1}&a_{11}&a_{12}&\dots &a_{1s}\\c_{2}&a_{21}&a_{22}&\dots &a_{2s}\\\vdots &\vdots &\vdots &\ddots &\vdots \\c_{s}&a_{s1}&a_{s2}&\dots &a_{ss}\\\hline &b_{1}&b_{2}&\dots &b_{s}\\\end{array}}

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