= Forward Euler stability for centered advection-diffusion
{c}
For $u_t=u_{xx}+\alpha u_x$ on the whole line, apply centered differences in space and the <Forward Euler method> in time. With $r=\Delta t/(\Delta x)^2$ and $c=\alpha\Delta t/\Delta x$, the amplification factor is
$$
G(\theta)=1-4r\sin^2(\theta/2)+ic\sin\theta.
$$
Putting $s=\sin^2(\theta/2)$ gives
$$
|G|^2-1=4s\{c^2-2r+(4r^2-c^2)s\}.
$$
For $\Delta t>0$, the affine expression in braces is nonpositive on $[0,1]$ exactly when $r\leq1/2$ and $c^2\leq2r$. Hence <von Neumann stability analysis> gives
$$
\Delta t\leq\frac{(\Delta x)^2}{2},
\qquad\alpha^2\Delta t\leq2.
$$
These conditions are weaker than requiring every stencil weight to be nonnegative; <stability> in the discrete <L2 norm> need not imply monotonicity.
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