= Adams-Bashforth stability for centered diffusion
{c}
{title2=$0\leq\Delta t/(\Delta x)^2\leq1/4$}
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 $P(-1)=2-2a\geq0$. 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 $\mu\leq1/4$. Under fixed $\mu$, the unscaled one-step defect is $-\Delta t^2u_{xxxx}/(12\mu)+O(\Delta t^3)$; dividing by the step changes its order to $O(\Delta t)$.
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