(a) Consider a method for numerically solving an ordinary differential equation (ODE) for an initial value problem, $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y})$. What does it mean for a method to converge over $t \in[0, T]$ where $T \in \mathbb{R}$ ? What is the definition of the order of a method?

(b) A general multistep method for the numerical solution of an ODE is

$$\sum_{l=0}^{s} \rho_{l} \mathbf{y}_{n+l}=h \sum_{l=0}^{s} \sigma_{l} \mathbf{f}\left(t_{n+l}, \mathbf{y}_{n+l}\right), \quad n=0,1, \ldots$$

where $s$ is a fixed positive integer. Show that this method is at least of order $p \geqslant 1$ if and only if

$$\sum_{l=0}^{s} \rho_{l}=0 \quad \text { and } \quad \sum_{l=0}^{s} l^{k} \rho_{l}=k \sum_{l=0}^{s} l^{k-1} \sigma_{l}, \quad k=1, \ldots, p$$

(c) State the Dahlquist equivalence theorem regarding the convergence of a multistep method.

(d) Consider the multistep method,

$$\mathbf{y}_{n+2}+\theta \mathbf{y}_{n+1}+a \mathbf{y}_{n}=h\left[\sigma_{0} \mathbf{f}\left(t_{n}, \mathbf{y}_{n}\right)+\sigma_{1} \mathbf{f}\left(t_{n+1}, \mathbf{y}_{n+1}\right)+\sigma_{2} \mathbf{f}\left(t_{n+2}, \mathbf{y}_{n+2}\right)\right]$$

Determine the values of $\sigma_{i}$ and $a$ (in terms of the real parameter $\theta$ ) such that the method is at least third order. For what values of $\theta$ does the method converge?