Adams-Bashforth stability for centered diffusion ($0\le \Delta t/(\Delta x{)}^2\le 1/4$)

For centered-space diffusion, a Fourier mode gives $z^2-(1-3a/2)z-a/2=0$, with $a=4\mu {\sin }^2(\theta /2)$. For $a>0$ the roots have opposite signs, the positive root lies in $(0,1)$, and the negative root lies in $[-1,0)$ exactly when $P(-1)=2-2a\ge 0$. At $a=0,1$ the unit-modulus roots are simple. Hence the root condition for a multistep method holds for every Fourier frequency exactly when $\mu \le 1/4$. Under fixed $\mu$, the unscaled one-step defect is $-\Delta t^2{u}_{xxxx}/(12\mu )+O(\Delta t^3)$; dividing by the step changes its order to $O(\Delta t)$.