 Central finite difference

A central finite difference uses sampled values symmetrically around an evaluation point. Common scaled derivative formulas areFor sufficiently smooth functions, they approximate $f^{\prime }(x)$ and $f^{\prime \prime }(x)$ with error $O(h^2)$. Taylor expansion explains the accuracy: symmetry cancels the unwanted even or odd powers. The leading errors are $h^2{f}^{(3)}(x)/6$ for $D_0$ and $h^2{f}^{(4)}(x)/12$ for $D_{xx}$ when the next derivatives permit these expansions.

At unit spacing and $x=0$, the second formula becomes $f(-1)-2f(0)+f(1)$, with finite difference coefficients $1,-2,1$. Symmetry is a local approximation property; it does not by itself establish stability of a time update or provide values beyond the boundary. Those require von Neumann stability analysis, an energy estimate, or an appropriate boundary closure of a difference scheme.