= Solution
Let $L_h=D_{xx}+\alpha D_x$, where the centered second-difference matrix $D_{xx}$ is real symmetric negative definite under homogeneous <Dirichlet boundary conditions>, and the centered first-difference matrix $D_x$ is real skew-symmetric. Hence
$$
\operatorname{Re}\langle\mathbf u,L_h\mathbf u\rangle
=\langle\mathbf u,D_{xx}\mathbf u\rangle\leq0.
$$
The semidiscrete <energy method> gives $\|\mathbf u(t)\|_2\leq\|\mathbf u(0)\|_2$ for every real $\alpha$, so the scheme is stable.
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