Runge–Kutta methods

Remarks

The 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. Usually $c_i = \sum_j a_{ij}$, and the method is consistent, of order at least $1$, exactly when $\sum_i b_i = 1$. Example: $s = 1$, $a_{11} = 0$, $b_1 = 1$, $c_1 = 0$ gives $y_{k+1} = y_k + h f(t_k, y_k)$, the Euler method. The 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 $4$. For $y' = \lambda y$ 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 $4$. Non-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.