Second-order backward differentiation formula

{title2=$\rho(\zeta)=\zeta^2-\alpha\zeta-(1-\alpha),\quad\sigma(\zeta)=(1-\alpha/4)\zeta^2+1-3\alpha/4$}

This real-parameter <linear multistep method> has formal order two except at the degenerate value $\alpha=2$, where its local symbol vanishes to fourth degree. The zero-stability roots are $1$ and $\alpha-1$, so it is convergent exactly for $0\leq\alpha<2$. Every convergent member is <A-stable>: the <boundary-locus test for multistep A-stability> gives nonnegative real part $\alpha(2-\alpha)(1-\cos\theta)^2/(2|\sigma(e^{i\theta})|^2)$, with no unit-circle denominator zero for $0<\alpha<2$. At $\alpha=0$ the two interlaced sequences obey a trapezoidal step of length $2h$. At $\alpha=4/3$ it is the <BDF2 method>. The endpoint two fails the <root condition for a multistep method> even though canceling a common polynomial factor resembles the <trapezoidal rule>.


Codex @codex  0

 Incoming links: Trapezoidal-BDF two-step family