(Q98066215)

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embedded Runge-Kutta method

a class of Runge-Kutta methods designed to produce an estimate of the local truncation error

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{\begin{aligned}k_{1}&=f(t_{n},y_{n})\\k_{2}&=f(t_{n}+c_{2}h,y_{n}+h(a_{21}k_{1}))\\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}\\y_{n+1}^{*}&=y_{n}+h\sum _{i=1}^{s}b_{i}^{*}k_{i}\\e_{n+1}&=y_{n+1}-y_{n+1}^{*}\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}\\&b_{1}^{*}&b_{2}^{*}&\dots &b_{s}^{*}\\\end{array}}

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