4.5. Stability exercises#
Answer the following exercises based on the content from this chapter. The solutions can be found in the appendices.
Exercise 4.1
Determine the stability function of the following Runge-Kutta method
\[\begin{split} \begin{array}{c|ccccc}
0 & & & & & \\
1/4 & 1/4 & & & & \\
1/2 & 1/2 & 0 & & & \\
3/4 & 0 & 1/2 & 1/4 & & \\
0 & 0 & 1/6 & -1/3 & 1/6 & \\ \hline
& -1 & 2/3 & -1/3 & 2/3 & 1
\end{array} \end{split}\]
Exercise 4.2
Determine the stability function of the following Runge-Kutta method. Is this an A-stable method?
\[\begin{split} \begin{array}{c|cc}
1/4 & 7/24 & -1/24\\
3/4 & 13/24 & 5/24\\ \hline
& 1/2 & 1/2
\end{array} \end{split}\]
Exercise 4.3
Plot the region of absolute stability for the following Runge-Kutta method.
\[\begin{split} \begin{array}{c|cc}
1/3 & 1/3 & 0\\
1 & 1 & 0\\ \hline
& 3/4 & 1/4
\end{array} \end{split}\]
Exercise 4.4
Calculate the stiffness ratio for the following system of ODEs.
\[\begin{align*}
y_1' &= -80.6 y_1 + 119.4 y_2,\\
y_2' &= 79.6 y_1 - 120.4 y_2.
\end{align*}\]
What are the maximum step lengths that the Euler method is stable for solving each equation?