An explicit $s$-staged Runge-Kutta method for an autonomous ODE $\dot y = L(y)$ can be written as $$ k_i = L\left(y_n + \tau\sum_{j=1}^{i-1} a_{ij} k_j \right)\\ y_{n+1} = y_n + \tau\sum_{i=1}^s b_i k_i. $$ It seems that there exist some methods for which all the coefficients $a_{ij}, b_{i}$ are positive for $j < i$, e.g.:
- $s = 1$ explicit Euler method $$ \begin{array}{c|c} &\\ \hline &1 \end{array} $$
- For $s = 2$ explicit trapezoid method $$ \begin{array}{c|cc} &&\\ 1&1\\ \hline &1/2 &1/2 \end{array} $$
- For $s = 3$ (could not find the name, but I've checked the order conditions) $$ \begin{array}{c|ccc} \\ 1&1\\ 1/2&1/4&1/4\\ \hline &1/6&1/6&2/3 \end{array}. $$
These methods are important for building TVD Runge-Kutta methods. It seems that there's no method with four stages of the fourth order with positive Butcher tableau, but I wonder if there are methods of fourth order and $s > 4$ satisfying the positiveness condition.
