= Solution
The <characteristic polynomials of a linear multistep method> are
$$
\rho(\zeta)=\zeta^2-(1+a)\zeta+a=(\zeta-1)(\zeta-a),\qquad
\sigma(\zeta)=\frac{1-3a}{2}\zeta+\frac{1+a}{2}\zeta^2.
$$
To determine formal order, substitute a smooth exact solution and expand about the first time level. The <exponential-symbol order criterion for a multistep method> collects precisely the same coefficients:
$$
\rho(e^z)-z\sigma(e^z)
=-\frac{1+5a}{12}z^3-\frac{3+11a}{24}z^4+O(z^5).
$$
The constant, linear and quadratic coefficients vanish for every $a$. The cubic coefficient vanishes only at $a=-1/5$, where the quartic coefficient is $-1/30\ne0$. Hence
$$
\boxed{p=3\text{ if }a=-\tfrac15,\qquad p=2\text{ otherwise}.}
$$
Here order means the exact-solution step residual is $O(h^{p+1})$. It is a formal consistency result, not a convergence assertion. In particular at $a=1$ both $\rho'(1)$ and $\sigma(1)$ vanish, and the double root at one destroys <zero-stability>; cancelling its common factor gives a different, first-order recurrence with an additional integration constant left unspecified by the original formula.
Back to article page
