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  "body": "\n## Definition\n\nLet $f \\colon \\R \\times \\R^n \\to \\R^n$, and consider the [[initial value problem|0FXDW992DS]] $y' = f(t, y)$, $y(t_0) = y_0$.\nLet $s \\ge 1$ be an integer and let $a_{ij}$, $b_i$, $c_i$ be real numbers for $i, j = 1, \\ldots, s$.\nThe *$s$-stage Runge–Kutta method* with these coefficients and step size $h \u003e 0$ computes approximations $y_k \\approx y(t_k)$ at $t_k = t_0 + k h$ by\n$$K_i = f\\Big( t_k + c_i h, \\; y_k + h \\sum_{j=1}^{s} a_{ij} K_j \\Big), \\quad i = 1, \\ldots, s, \\qquad y_{k+1} = y_k + h \\sum_{i=1}^{s} b_i K_i.$$\nThe method is *explicit* if $a_{ij} = 0$ for all $j \\ge i$, and *implicit* otherwise.\n\n## Remarks\n\nThe coefficients are written in the Butcher tableau with the $c_i$ in a column, the matrix $(a_{ij})$ beside it, and the $b_i$ as a bottom row.\nUsually $c_i = \\sum_j a_{ij}$, and the method is consistent, of order at least $1$, exactly when $\\sum_i b_i = 1$.\nExample: $s = 1$, $a_{11} = 0$, $b_1 = 1$, $c_1 = 0$ gives $y_{k+1} = y_k + h f(t_k, y_k)$, the [[Euler method|0JARTXX4XP]].\nThe classical method has $s = 4$, $c = (0, \\tfrac12, \\tfrac12, 1)$, $a_{21} = a_{32} = \\tfrac12$, $a_{43} = 1$, the other $a_{ij} = 0$, and $b = (\\tfrac16, \\tfrac13, \\tfrac13, \\tfrac16)$; it has order $4$.\nFor $y' = \\lambda y$ it gives $y_{k+1} = \\left( 1 + z + \\frac{z^2}{2} + \\frac{z^3}{6} + \\frac{z^4}{24} \\right) y_k$ with $z = h \\lambda$, the Taylor polynomial of $e^z$ of degree $4$.\nNon-example: the two-step formula $y_{k+2} = y_k + 2 h f(t_{k+1}, y_{k+1})$ is not a Runge–Kutta method, since $y_{k+2}$ depends on two earlier values and not only on $y_{k+1}$; it is a linear multistep method.\n",
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