Discretization

Discretization is the process of converting a continuous model or variable into a discrete representation, typically by replacing continuous domains, derivatives, or functions with their discrete analogs defined on a grid, lattice, or mesh. It enables numerical computation and simulation on digital computers and is central to numerical analysis and computer modeling.

In numerical analysis, discretization applies to differential equations and integral equations. Time (temporal) and space (spatial) discretization separate the independent variable into steps. Finite difference methods approximate derivatives by differences on a grid; finite element methods use piecewise polynomial basis functions on a mesh; finite volume methods conserve fluxes across control volumes. Spectral methods approximate solutions by global basis functions. Mesh generation and adaptivity (refinement of the grid where needed) are important practical aspects.

Discretization introduces errors, collectively called discretization or truncation errors. The size of the step or mesh determines the order of accuracy; smaller steps increase accuracy but raise computational cost. Stability and convergence describe whether the discrete solution behaves well and approaches the continuous solution as the discretization is refined.

In signal processing and data acquisition, discretization also includes sampling in time and quantization of amplitude, converting continuous signals into discrete-time, discrete-amplitude representations. The Nyquist-Shannon sampling theorem guides how frequently to sample to avoid aliasing.

Applications include solving differential equations in physics and engineering, simulating fluid flow and structural analysis, computer graphics, and data-driven modeling and numerical integration.