centraldifference

Central difference is a symmetric finite difference scheme used to approximate derivatives by evaluating a function at points equidistant from the target location. It is commonly described as the symmetric difference quotient.

For a function f defined on a grid with spacing h, the simplest central difference estimate of the first derivative at x is

f′(x) ≈ [f(x + h) − f(x − h)] / (2h).

This estimator is second-order accurate, with a leading error term proportional to h^2.

The central difference for the second derivative is

f″(x) ≈ [f(x + h) − 2f(x) + f(x − h)] / h^2,

also second-order accurate, with a leading error term proportional to h^2.

Compared with forward or backward differences, central differences are more accurate for the same grid spacing and yield symmetric error terms, making them well suited for smooth functions. They rely on function values on both sides of the point, which can complicate use near boundaries or on irregular grids.

Common applications include numerical differentiation and the discretization of differential equations via finite difference methods. Central differences are widely used to approximate spatial derivatives in solving boundary value problems and partial differential equations, such as Poisson or diffusion equations, and to implement second-order time-stepping schemes in certain second-order ODE contexts.

Limitations include reduced applicability near domain boundaries without one-sided formulas, and diminished accuracy for non-smooth or noisy data where the symmetric stencil amplifies irregularities.