= Fourth-order centered second derivative
{title2=$D_{4,h}u_m=(-u_{m-2}+16u_{m-1}-30u_m+16u_{m+1}-u_{m+2})/(12h^2)$}
This symmetric <finite difference method> approximates a second derivative with $D_{4,h}u=u''-h^4u^{(6)}/90+O(h^6)$ for sufficiently <smooth> functions. Matching the constant, second-derivative and fourth-derivative coefficients of a symmetric five-node <Taylor expansion> determines the displayed weights uniquely; its sixth-derivative coefficient is nonzero. The <Fourier symbol> is $-4h^{-2}\sin^2(\theta/2)[1+\sin^2(\theta/2)/3]\le0$. Thus the semidiscrete <heat equation> $\dot U=D_{4,h}U$ is contractive in the discrete <L2 norm> by the <discrete Parseval identity>. This says nothing by itself about stability of a subsequently chosen time update.
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