Past exam of the numerical analysis course of the University of Oxford / 2021 / 3 / c / i / Solution

No. A necessary condition for consistency of a numerical method is reproduction of the constant solution $y(x)=1$ of the ordinary differential equation $y^{\prime }=0$. In a linear multistep method this givesBut the reverse triangle inequality gives, under the strict coefficient inequality,Thus the necessary order conditions for a linear multistep method already fail at order zero, regardless of the values of the ${\beta }_j$.