Every member of the family has order at least three, so it is <consistency of a numerical method>[consistent]. By the <Dahlquist equivalence theorem>, convergence is therefore equivalent to <zero-stability>, which is characterized by the <root condition for a multistep method>.

The roots of $\rho$ are $1$ and $a$. They lie in the closed unit disk exactly when $|a|\leq1$. At $a=1$, however, the unit root $1$ is repeated, whereas at $a=-1$ the distinct unit roots $1$ and $-1$ are both simple. Consequently
$$
\boxed{\text{the method is convergent exactly for }-1\leq a<1.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 341 / 2 / b / Solution

A numerical method is consistent when its local defect tends to zero at the rate required to approximate the differential equation. Stability then determines whether these local defects accumulate into a convergent global approximation.


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 Incoming links: Consistency of a numerical method