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I know that for the RK2 method to be consistent we must have $a + b = 1$ in the following equation

$$\begin{aligned} y_{i+1} &= y_i + h(ak_1 + bk_2)\\ k_1 &= f(x_i, y_i)\\ k_2 &= f(x_i + \alpha h, y_i + \beta hk_1) \end{aligned} $$

However, for the general $s$-stage Runge-Kutta method, what conditions must one have on the terms in the Butcher tableau in order to ensure consistency?

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Are you interested in general explicit (as the given one is) RK-methods or in implicit ones? – martini Nov 19 '12 at 8:02
I am interested in general explicit RK-methods, not implicit ones. – sonicboom Nov 19 '12 at 8:32
up vote 1 down vote accepted

We consider the explicit RK-method \begin{align*} y_{n+1} &= y_n + h \sum_{i=1}^s b_ik_i\\ k_i &= f\left(x_n + c_ih, y_n + h \sum_{j=1}^{i-1}a_{ij}k_j\right) \end{align*} with Butcher tableau $$ \begin{array}{c|cccc} c_1 & 0 & \cdots & 0 & 0\\ c_2 & a_{21} & \cdots & 0 & 0 \\ \vdots & & \vdots \\ c_s & a_{s1} & \cdots & a_{s,s-1} & 0 \\ \hline & b_1 & \cdots & b_{s-1} & b_s \end{array} $$ This is constistent (just the same as in the $s=2$ case) iff $\sum_{i=1}^s b_i = 1$. To see this, just note that $k_i \to f(x_n, y_n)$ as $h \to 0$ and hence $$ \sum_{i=1}^n b_i k_i \to \sum_{i=1}^n b_i f(x_n, y_n), \qquad h \to 0. $$

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