A numerical technique that approximates derivatives in differential equations by algebraic difference quotients on a discrete grid, producing a system of algebraic equations whose solutions approximate the continuous problem.

Principle

Replace differential operators by local discrete stencils (forward, backward, central differences) whose truncation error determines the scheme's order; combine consistency and stability to obtain convergence of the discrete solution to the continuous one.

Demonstration

For the one-dimensional Poisson equation u''(x)=f(x) on [0,1], a second-order central difference yields (u_{i+1}-2u_i+u_{i-1})/h^2 = f_i, producing a tridiagonal linear system approximating the continuous boundary value problem.

Misapplication

Using an unstable time-stepping finite difference scheme for a stiff parabolic problem or applying low-order stencils on coarse grids expecting high accuracy leads to spurious oscillations or convergence failure.

Consequence

Transforms differential boundary value or initial value problems into linear or nonlinear algebraic systems amenable to direct or iterative solvers; error behavior follows the interplay of discretization (consistency), stability, and mesh resolution.

Reversal

Spectral or global-basis methods replace local stencils by global expansions and invert the usual locality–accuracy tradeoff: they can be more accurate for smooth solutions but less flexible for complex geometries than finite differences.

Boundary

Most natural on structured grids and for problems with regular domains and boundary conditions; irregular geometries, variable coefficients with high anisotropy, or requirements for weak formulations may favor other discretizations.

Semantic Tension

Often contrasted with finite element methods: finite differences discretize strong differential operators via pointwise stencils, while finite elements use weak formulations and basis functions, producing different assembly and convergence properties.

Synthesis

A local-stencil discretization approach converting derivatives into algebraic differences on a grid; when consistent and stable, it yields convergent discrete approximations to differential problems suitable for numerical solvers.