3.4. Implicit Runge-Kutta methods exercises

Answer the following exercises based on the content from this chapter. The solutions can be found in the appendices.

Exercise 3.1

Determine the order of the DIRK shown below.

\[\begin{split} \begin{array}{c|cc} 1/4 & 1/4 \\ 3/4 & 1/2 & 1/4 \\ \hline & 1/2 & 1/2 \end{array} \end{split}\]

Exercise 3.2

Derive a third-order Radau IIA method.

Exercise 3.3

Calculate the first step of the third-order Radau IIA method derived in Exercise 3.2 to solve the following initial value problem using a step length of \(h=0.4\) and a accuracy tolerance of \(tol = 10^{-4}\)

\[\begin{align*} y' =t - y, \qquad t \in [0,2], \qquad y(0) = 1. \end{align*}\]

Exercise 3.4

The exact solution to the IVP in Exercise 3.3 is \(y = t + 2e^{-t} - 1\). Write a program to this initial value problem over the full domain, produce a table comparing the numerical and exact solutions and plot the numerical solutions and exact solutions on the same set of axes.