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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.

A finite difference method approximates a differential equation by replacing derivatives with weighted combinations of nearby sampled function values. Finite difference coefficients are chosen by matching low-degree polynomials or a Taylor expansion. The resulting unknowns live on a grid, where the differential equation becomes a system of algebraic update or boundary equations.

For the one-dimensional heat equation $u_t=\nu u_{xx}$ with $\nu >0$, a forward Euler method in time and a second-order central difference in space giveFor $0\le r\le 1/2$, the coefficients are nonnegative and sum to one, so each new value is a convex combination of neighboring values. This yields a maximum norm contraction for compatible homogeneous boundaries.^[ref]

Consistency of a numerical method measures how accurately the grid equation reproduces a smooth exact solution; stability of a numerical method controls propagation of perturbations. Both matter for numerical convergence. A correct interior formula also needs a compatible boundary closure of a difference scheme and suitable initial conditions.