Calculations - absolute stability

[evinda]

Hello! (Wave) The following Runge-Kutta method is given.

$$ \begin{array}{c|ccccc}
\tau_1 =0 & a_{11}=0 & a_{12} = 0\\
\tau_2 =\frac{5}{2} & a_{21} = \frac{5}{2} & a_{22} = 0\\
\hline
& b_1 = \frac{4}{5} & b_2 = \frac{1}{5} & \
\end{array} $$

I have to determine the order of accuracy and I found that it is $2$.


Then if the method is applied at the system $\\y_1'=-80y_1+20y_2 \\
y_2'=20y_1-80y_2$ what $h$ should we take so that the calculations get done with absolute stability?

How can we find such an $ h$?
I found that the region of absolute stability is $S=\{ z \in \mathbb{C}: |\frac{z^2}{2}+z+1|<1\}$. Does this help?

 

[I like Serena]

Hi! (Mmm)

What do your notes say absolute stability is exactly? And how to find it? (Wondering)

Wiki says that the stability function $r$ is:
$$r(z) = 1 + z b^T (I-zA)^{-1} e = \frac{\det(I-zA+zeb^T)}{\det(I-zA)}$$
where $e$ stands for the vector of ones, and $b^T$ is the bottom row of the tableau. (Thinking)