Finitedifference

Finite difference methods are a family of numerical techniques for approximating derivatives and solving differential equations by replacing continuous operators with finite difference quotients on a grid. The basic idea is to sample the unknown function at a set of grid points spaced by h and express derivatives as linear combinations of these sampled values. In one dimension, the forward difference for the first derivative is f'(x) ≈ (f(x+h) - f(x))/h, the backward difference is f'(x) ≈ (f(x) - f(x-h))/h, and the central difference is f'(x) ≈ (f(x+h) - f(x-h))/(2h). The second derivative can be approximated by f''(x) ≈ (f(x+h) - 2f(x) + f(x-h))/h^2. Central differences typically have higher accuracy, with truncation error O(h^2) versus O(h) for forward/backward formulas.

In multiple dimensions, similar stencils are used. For example, the discrete Laplacian in two dimensions on a square grid with spacing h is ∆_h u_{i,j} = (u_{i+1,j} - 2u_{i,j} + u_{i-1,j})/h^2 + (u_{i,j+1} - 2u_{i,j} + u_{i,j-1})/h^2, which appears in finite difference approximations to Poisson’s equation ∆u = f. Boundary conditions (Dirichlet, Neumann) are imposed on boundary grid points. The result is a sparse linear system that can be solved by direct or iterative methods.

Finite differences also apply to time-dependent problems via explicit and implicit schemes, such as forward Euler y_{n+1} = y_n + h f(t_n, y_n) or Crank–Nicolson, with stability considerations (for example, CFL conditions) guiding the choice of time step. Finite difference methods are valued for their simplicity, locality, and broad applicability to ODEs and PDEs.