= Solution
Write the <method of lines>[semidiscrete system] as $\mathbf u'=L_h\mathbf u+\alpha D_h\mathbf u$. With zero boundary values, the centered second-difference matrix $L_h$ is symmetric negative definite and the centered first-difference matrix $D_h$ is <skew-symmetric matrix>[skew-symmetric]. Therefore
$$
\frac12\frac d{dt}\|\mathbf u(t)\|_2^2
=\mathbf u^TL_h\mathbf u+\alpha\mathbf u^TD_h\mathbf u
=\mathbf u^TL_h\mathbf u\leq0.
$$
This is a mesh-uniform stability estimate for the <centered convection-diffusion semidiscretization>, valid for every real $\alpha$.
Both centered differences have local spatial error $O(h^2)$ for a sufficiently smooth solution. Stability plus consistency gives convergence, equivalently by the semidiscrete form of the <Lax equivalence theorem>. Thus
$$
\boxed{\text{the semidiscretization converges with spatial order two for every fixed }\alpha\in\mathbb R.}
$$
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