Paper 1, Section I, 6C\mathbf{6 C}

Numerical Analysis
Part IB, 2014

(i) A general multistep method for the numerical approximation to the scalar differential equation y=f(t,y)y^{\prime}=f(t, y) is given by

=0sρyn+=h=0sσfn+,n=0,1,\sum_{\ell=0}^{s} \rho_{\ell} y_{n+\ell}=h \sum_{\ell=0}^{s} \sigma_{\ell} f_{n+\ell}, \quad n=0,1, \ldots

where fn+=f(tn+,yn+)f_{n+\ell}=f\left(t_{n+\ell}, y_{n+\ell}\right). Show that this method is of order p1p \geqslant 1 if and only if

ρ(ez)zσ(ez)=O(zp+1) as z0\rho\left(\mathrm{e}^{z}\right)-z \sigma\left(\mathrm{e}^{z}\right)=\mathcal{O}\left(z^{p+1}\right) \quad \text { as } \quad z \rightarrow 0

where

ρ(w)==0sρw and σ(w)==0sσw\rho(w)=\sum_{\ell=0}^{s} \rho_{\ell} w^{\ell} \quad \text { and } \quad \sigma(w)=\sum_{\ell=0}^{s} \sigma_{\ell} w^{\ell}

(ii) A particular three-step implicit method is given by

yn+3+(a1)yn+1ayn=h(fn+3+=02σfn+)y_{n+3}+(a-1) y_{n+1}-a y_{n}=h\left(f_{n+3}+\sum_{\ell=0}^{2} \sigma_{\ell} f_{n+\ell}\right)

where the σ\sigma_{\ell} are chosen to make the method third order. [The σ\sigma_{\ell} need not be found.] For what values of aa is the method convergent?