I am failing to understand how to write a Runge-Kutta method, such as the example below into a Butcher tableau.
A uniform s-stage Runge-Kutta method for the scalar initial-value problem
$$\frac{du}{dt}=f(t,u)$$ $$u(0)=u_0$$
is given by the following:
- Let $U^0:=u(0)$
- For each $i = 1,\dots, s$ compute $p_i$ such that
$$\quad\quad\quad\quad\quad\quad\quad p_i=U^n+\tau\sum_{j=1}^s a_{ij}f(t_n+\tau c_j, p_j)\quad\quad\quad\quad\quad(1)$$
- Finally, set
$$\quad\quad\quad\quad\quad\quad\quad U^{n+1}=U^n+\tau\sum_{i=1}^s b_{i}f(t_n+c_i\tau, p_i) \quad\quad\quad\quad(2)$$
Write the method $(1)–(2)$ in the form of a Butcher tableau
In a first step to get the perhaps better recognizable form of a Runge-Kutta method, write $$ k_i=\tau f(t_n+\tau c_i, p_i)=\tau f(t_n+c_i\tau, U_n+\sum_{j=1}^s a_{ij}k_j) $$ Then also $$ U_{n+1}=U_n+\sum_{i=1}^s b_ik_i. $$ The Butcher tableau is then, with the already conventionally named coefficients, $$ \begin{array}{c|cccc} c_1&a_{11}&\dots&a_{1s}\\ \vdots&\vdots&&\vdots\\ c_s&a_{s1}&\dots&a_{ss}\\ \hline &b_1&\dots&b_s \end{array} $$