Requirement(s) for consistency of Runga Kutta methods?

Question

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?

Are you interested in general explicit (as the given one is) RK-methods or in implicit ones? — martini, Nov 19 '12 at 08:02
I am interested in general explicit RK-methods, not implicit ones. — sonicboom, Nov 19 '12 at 08:32

score 2 · Accepted Answer · answered Nov 19 '12 at 09:10

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 $$ \sum_{i=1}^n b_i k_i \to \sum_{i=1}^n b_i f(x_n, y_n), \qquad h \to 0. $$

Requirement(s) for consistency of Runga Kutta methods?

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