Central finite difference

The centered unit-step approximation to a second derivative is

The weights $1,-2,1$ are finite difference coefficients for this centered second-derivative approximation.

On a uniform mesh $x_n=a+nh$, the centered second finite difference is exact for every quadratic polynomial. In particular, for $y^{\prime \prime }=c$ with endpoint boundary conditions, the unique discrete solution equals the exact quadratic solution at every mesh point. Subtracting consecutive first differences reduces the equation to $d_{n+1}-d_n=c{h}^2$, so $y_n=\frac{1}{2}c{h}^{2}n(n-1)+d_{1}n+y_0$. The second endpoint determines $d_1$. Uniqueness follows because any difference of two discrete solutions has zero second finite difference, hence is linear in the mesh index and vanishes at both endpoints. Exact nodal values in this example do not imply that a general finite difference method is exact for other ordinary differential equations.

This symmetric finite difference method approximates a second derivative with $D_{4,h}u=u^{\prime \prime }-h^4{u}^{(6)}/90+O(h^6)$ 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 $-4h^{-2}{\sin }^2(\theta /2)[1+{\sin }^2(\theta /2)/3]\le 0$. 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.

For $f\in C^3[x-h,x+h]$, the Peano kernel for the error of the central secant slope has one sign. Integrating its absolute value gives the displayed constant $h^2/6$, and a cubic polynomial attains it. At $x=1$, $h=1$, the unnormalized order-two kernel is $-{\theta }^2/2$ on $[0,1]$ and $-(2-\theta {)}^2/2$ on $[1,2]$; the integral representation includes the factor $1/2!$.

Suppose $|f^{\prime \prime \prime }|\le M$ and each sampled value of $f$ has absolute error at most $\delta$. For a central finite difference with $h>0$, the triangle inequality and the sharp central-secant derivative error giveFor $M,\delta >0$, the derivative of this bound is $Mh/3-\delta /h^2$. It changes sign at $h_{\ast }=(3\delta /M{)}^{1/3}$, proving the unique minimum, of value $(3\delta {)}^{2/3}{M}^{1/3}/2$. This balance explains why reducing the step indefinitely amplifies sample absolute error even as truncation error decreases.

Forward time with centered advection has amplification magnitude greater than one at nonzero modes. Under a fixed nonzero Courant number, its powers blow up as the time step vanishes. If instead $\Delta t/h^2$ stays bounded, then $\log |G|\le c^2\Delta t^2/(2h^2)$, so $|G{|}^n\le \exp (c^{2}T\Delta t/(2h^2))$ is uniformly bounded on fixed time intervals. This severe parabolic refinement supplies stability without contraction; it does not validate the usual hyperbolic-step choice.

On a uniform grid of spacing $h$, the second-order central differenceapproximates $f^{\prime \prime }(x)$ with error $O(h^2)$ for a sufficiently smooth function.