= Fourier stability threshold for a mixed-width heat scheme
{c}
{title2=$\mu(1-2\theta)^2\leq2(1-\theta)$}
Consider an implicit centered heat stencil of width $\Delta x$ weighted by $\theta$, combined with an explicit centered stencil of width $2\Delta x$ weighted by $1-\theta$. With $\mu=\kappa\Delta t/(\Delta x)^2$, its <amplification factor> is $G(\xi)=[1-\mu(1-\theta)\sin^2\xi]/[1+4\mu\theta\sin^2(\xi/2)]$. For $0\leq\theta<1/2$, contraction in the discrete $\ell^2$ norm holds exactly under the displayed condition. It is unconditional for $1/2\leq\theta\leq1$. Put $v=\sin^2(\xi/2)$: the condition $G\geq-1$ reduces to $4\mu v[(1-2\theta)-(1-\theta)v]\leq2$. Maximizing this concave quadratic proves the threshold; the <Parseval identity> converts the multiplier bound into contraction.
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