Discretisere

Discretisere (to discretize) is a Danish verb used in mathematics, computer science and engineering to describe the process of turning a continuous problem or quantity into a discrete one. The concept is central to turning mathematical models into forms suitable for numerical computation.

The core idea is to replace a continuous domain—such as time, space, or a function—with a finite set of points, cells, or basis functions. This yields a system with a finite number of degrees of freedom, enabling numerical computation.

Discretization translates differential equations into algebraic equations, or continuous operators into matrices. Spatial discretization uses meshes or grids; temporal discretization uses time steps.

Common methods include finite difference method (grid points, approximating derivatives by differences), finite element method (piecewise polynomial basis on a mesh), finite volume method (conservation-law based on fluxes across control volumes), and spectral methods (global basis functions). Time-stepping schemes include explicit and implicit methods, such as Euler, Runge-Kutta, and Crank-Nicolson.

Discretization introduces errors generally called discretization error, related to step size or mesh resolution. Key concerns are consistency, stability, and convergence (Lax equivalence theorem for linear schemes). Adaptive methods refine the mesh where the error is large.

Applications span solving ordinary and partial differential equations, physical simulations, computer graphics, signal processing (sampling), and, in data analysis, discretization of continuous variables for categorization or binning.