 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. HenceThe semidiscrete energy method gives $\Vert \mathbf{u}(t){\Vert }_2\le \Vert \mathbf{u}(0){\Vert }_2$ for every real $\alpha$, so the scheme is stable.