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
$$
\boxed{\text{the semidiscretization converges with spatial order two for every fixed }\alpha\in\mathbb R.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 341 / 4 / b / Solution

With homogeneous <Dirichlet boundary conditions>, centered differences turn one-dimensional diffusion into a symmetric negative-definite matrix and constant advection into a skew-symmetric matrix. Their sum dissipates the discrete <Euclidean norm>, giving mesh-uniform energy stability for every fixed advection coefficient.

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 Incoming links: Centered convection-diffusion semidiscretization