Butcher tableau in Runge Kutta methods

Question

I'm studying RK methods using Butcher tableau.
I do not understand what this means. Can anyone explain what is the notation?
RK Butcher tableau matrix It seems like matrices multiplication but MM does not look like that.
Thanks in advance.

Answer 1

It is just a scheme to hold the parameters of the method, just as a general matrix is a scheme to hold the coefficients of a linear map. \begin{array}{c|ccc} \color{red}{c_1}&\color{green}{a_{11}}&\cdots&\color{green}{a_{1s}} \\ \vdots&\vdots&&\vdots\\ \color{red}{c_s}&\color{green}{a_{s1}}&\cdots&\color{green}{a_{ss}} \\ \hline &\color{blue}{b_1}&\cdots&\color{blue}{b_s} \end{array}

If you have seen a general Runge-Kutta method, you should have seen these parameters applied in their correct place, \begin{alignat}{1} \vec k_i&=\vec f\bigl(t+\color{red}{c_i}h, &\vec y&+\sum_{j=1}^s \color{green}{a_{ij}}\vec k_jh\bigr),~~~ i=1,...,s,\\ \vec y_{+1}&=&\vec y&+\sum_{i=1}^s \color{blue}{b_i}\vec k_ih \end{alignat} So in some sense there is some kind of matrix-vector multiplication done with the coefficients $a_{ij}$.