Chapter 3

Restricting your attention to scalar autonomous equations \(y^{\prime}=f(y)\), prove that the ERK method with tableau \begin{tabular}{c|cccc} 0 & & & & \\ \(\frac{1}{2}\) & \(\frac{1}{2}\) & & & \\ \(\frac{1}{2}\) & 0 & \(\frac{1}{2}\) & & \\ 1 & 0 & 0 & 1 & \\ \hline & \(\frac{1}{6}\) & \(\frac{1}{3}\) & \(\frac{1}{3}\) & \(\frac{1}{6}\) \end{tabular} is of order 4 .

Suppose that a \(\nu\)-stage ERK method of order \(\nu\) is applied to the linear scalar equation \(y^{\prime}=\lambda y\). Prove that $$ y_{n}=\left[\sum_{k=0}^{\nu} \frac{1}{k !}(h \lambda)^{k}\right]^{n} y_{0}, \quad n=0,1, \ldots $$

Write the theta method, (1.13), as a Runge-Kutta method.