eindigdifference

Eindigdifference, in English often referred to as finite difference, is a concept in numerical analysis used to approximate derivatives and to discretize differential equations. It replaces a continuous function by values at a finite set of grid points and expresses derivatives as differences between these values.

For a function f defined on a grid x_i with uniform spacing h, several basic differences are used. The forward difference is Δ_h f(x_i) = f(x_{i+1}) - f(x_i), and the backward difference is ∇_h f(x_i) = f(x_i) - f(x_{i-1}). The central difference is δ_h f(x_i) = (f(x_{i+1}) - f(x_{i-1}))/2. The first derivative can be approximated by these differences, with truncation errors that are O(h) for forward and backward differences and O(h^2) for the central difference. The second difference Δ_h^2 f(x_i) = f(x_{i+1}) - 2f(x_i) + f(x_{i-1}) approximates the second derivative f''(x_i).

Applications of eindigdifference are widespread in numerical differentiation and in finite difference methods for solving differential equations. By discretizing a domain into a grid, differential operators are replaced by difference operators, enabling the numerical solution of boundary value problems such as Poisson’s or diffusion equations. Central differences are commonly used because they typically yield higher accuracy (second-order) for a given grid spacing.

The accuracy of eindigdifference relies on the grid spacing h and the smoothness of the function, as explained by Taylor expansions. Higher-order finite-difference formulas can further improve precision. For example, with f(x) = x^2, the central difference for the first derivative yields exactly f'(x) = 2x.