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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.

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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

$$ \begin{array}{c|ccccc} 0 & & & & & \\ 2/5 & 2/5 & & & & \\ 3/5 & 1/10 & 1/2 & & & \\ 1/2 & 1/16 & 1/16 & 3/8 & & \\ 1 & 1/10 & 1/10 & 4/15 & 8/15 & \\ \hline & 5/32 & 25/96 & 25/96 & 1/6 & 5/32 \end{array} $$

It was found in:
J. F. B. M. Kraaijevanger. Contractivity of Runge-Kutta methods. BIT, 31:482–528, 1991.

You can find further methods and more details on TVD methods (also known as SSP methods) in the book "STRONG STABILITY PRESERVING RUNGE–KUTTA AND MULTISTEP TIME DISCRETIZATIONS" by Gottlieb, Ketcheson, Shu.

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