From Runge-Kutta Butcher tableau to general linear methods matrices?

Question

I am trying to understand how the relationship between Butcher tables for Runge-Kutta methods and their generalization to general linear methods matrices (by Butcher also).

• Runge-Kutta methods can be specified by the nodes $c_i$, the weights $b_j$ and the coefficients $a_{ij}$ under the form of Butcher tables (see Wikipedia here) (as well as $b_j^*$ for adaptive methods)
• General linear methods can be specified with a table of four matrices $A$, $B$, $V$, $U$ (see Wikipedia here)

Since Runge-Kutta methods can be seen as a subclass of general linear methods, I am wondering how $c_i$, $b_j$, $b_j^*$, $a_{ij}$ relate to $A$, $B$, $V$, $U$ (in particular $c_i$) (how to express the first ones in terms of the second ones, and vice-versa).

Answer 1

Given a Runge-Kutta method with the tableau

$$ \begin{array}{c|c} c & A \\ \hline & b^T \end{array} = \begin{array}{c|ccc} c_1 & a_{1,1} & \dots & a_{1,s} \\ \vdots & \vdots & \ddots & \vdots \\ c_s & a_{s,1} & \dots & a_{s,s} \\ \hline & b_1 & \dots & b_s \end{array} $$

it is equivalent to a GLM with the tableau

$$ \begin{array}{c|c|c} c & A & U \\ \hline & B & V \end{array} = \begin{array}{c|c|c} c & A & e \\ \hline & b^T & 1 \end{array} = \begin{array}{c|ccc|c} c_1 & a_{1,1} & \dots & a_{1,s} & 1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ c_s & a_{s,1} & \dots & a_{s,s} & 1 \\ \hline & b_{1} & \dots & b_{s} & 1 \end{array} $$

The Taylor series weights for the external stages are

$$ W = [q_0, q_1, \dots, q_k] = [1, 0, \dots, 0] $$

since $y_1^{[n]} = y_n$. Note that for a GLM with stage order one, $c = A e + U q_1$. This is discussed in "General Linear Methods for Ordinary Differential Equations" section 2, for example.

Embedded methods for GLMs are not as standardized as they are for Runge-Kutta schemes.