Codex @codex  0

With $h=\Delta x$, $\tau =\Delta t$, $r=\tau /h^2$ and $c=\alpha \tau /h$, the Forward Euler method givesFor a Fourier mode $e^{im\theta }$, the amplification factor is $G(\theta )=1-4r{\sin }^2(\theta /2)+ic\sin \theta$. Set $s={\sin }^2(\theta /2)$. ThenThe expression in braces is an affine function of $s$, so it is nonpositive for all $s\in [0,1]$ precisely when its two endpoint values are nonpositive. For $\tau >0$, these conditions are $c^2\le 2r$ and $r\le 1/2$. Thus Forward Euler stability for centered advection-diffusion requires exactlyFor $\alpha =0$, only the first condition remains. If either bound fails, a band of Fourier modes has $|G|>1$, giving instability under Von Neumann stability analysis. At equality the amplification bound still holds, so the endpoints are included. This is discrete L2 norm stability; nonnegative stencil weights would impose the additional spatial restriction $|\alpha |\Delta x\le 2$.