Finite difference method

A numerical method that approximates derivatives by combining function values at discrete grid points. It turns differential equations into algebraic equations that can be solved computationally.

Forward difference: A finite difference approximation that estimates a derivative from a point and a nearby point to its right. This one-sided formula is the simplest direct approximation to a first derivative.
Finite difference method for the heat equation: A grid-based discretization of the heat equation that advances or solves temperature values numerically. It is a standard example of approximating a time derivative and a second spatial derivative.
Finite element method: A numerical method that approximates solutions using local basis functions over a mesh of elements. It builds approximations from element-wise functions rather than direct derivative differences at grid points.
Taylor series: A representation of a sufficiently smooth function as a power series around a point. Taylor expansions derive difference formulas and expose their leading approximation errors.
FTCS scheme: A forward-time, centered-space finite difference scheme for time-dependent differential equations. This familiar explicit scheme demonstrates that a simple formula may have restrictive stability limits.
Backward difference: A finite difference approximation that estimates a derivative from a point and a nearby point to its left. Its one-sided stencil uses past or leftward values, useful when information flows in one direction.
Finite difference method for the wave equation: A grid-based scheme for approximating solutions of the wave equation over space and time. Its behavior shows how discretization can affect propagation speed and stability.
Finite volume method: A numerical method that enforces conservation by balancing fluxes across the boundaries of discrete control volumes. Its conservation-based formulation contrasts with direct pointwise derivative approximation.
Partial differential equation: An equation involving an unknown function and its partial derivatives with respect to multiple variables. Finite difference methods commonly replace these derivatives with grid-based formulas.
Crank–Nicolson method: An implicit time-stepping method that averages spatial operators at the current and next time levels. It offers a contrasting time discretization often paired with finite differences for diffusion problems.

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