# Runge-Kutta methods with strictly positive Butcher tableau

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.

Yes, there are methods of fourth order and $s>4$ satisfying the positiveness condition. For example, the following 5-stage RK method has order p=4
$$\begin{array}{c|ccccc} 0 & & & & & \\ 2/5 & 2/5 & & & & \\ 3/5 & 1/10 & 1/2 & & & \\ 1/2 & 1/16 & 1/16 & 3/8 & & \\ 1 & 1/10 & 1/10 & 4/15 & 8/15 & \\ \hline & 5/32 & 25/96 & 25/96 & 1/6 & 5/32 \end{array}$$