= Solution
Use the discrete inner product $\langle U,V\rangle_h=h\sum_{m=1}^{J-1}\overline U_mV_m$, with endpoints $U_0=U_J=0$. The centered first-difference <matrix> $D_x$ is skew-Hermitian, while <summation by parts> gives
$$
\operatorname{Re}\langle U,D_{xx}U\rangle_h=-\frac1h\sum_{m=0}^{J-1}|U_{m+1}-U_m|^2.
$$
Since $\alpha$ is real, the centered advection contribution has zero real part. Therefore
$$
\boxed{\frac{d}{dt}\|U\|_h^2=-\frac2h\sum_{m=0}^{J-1}|U_{m+1}-U_m|^2\leq0.}
$$
This proves the <energy contraction for centered drift-diffusion>, and hence <stability> for every fixed real $\alpha$ and every mesh width, in the <discrete L2 norm>. The extra inequality $|\alpha|h\leq2$ would make the off-diagonal coefficients nonnegative and support a discrete maximum principle, but is not needed for this energy proof. No time-stepping method is involved here.
Back to article page
