Glossary term
Discretization
The conversion of a continuous model, geometry, domain, signal, or equation into a finite set of elements, cells, samples, nodes, or time steps.
Definition
methodThe conversion of a continuous model, geometry, domain, signal, or equation into a finite set of elements, cells, samples, nodes, or time steps.
Discretization is the bridge between continuous mathematics and numerical computation. It introduces approximation error, stability constraints, resolution limits, and modelling choices that must be checked before simulation or sampled data can be trusted.
Discretization converts a continuous problem into a finite representation that a computer, calculation procedure, or digital system can handle. A continuous geometry may become a mesh. A differential equation may become algebraic equations. A continuous-time signal may become samples. A continuous design variable may become a finite grid or set of choices.
Engineering role
Discretization is central to finite-element analysis, computational fluid dynamics, heat transfer, structural dynamics, digital control, signal processing, optimization, and numerical solution of differential equations. It determines what length scales, time scales, gradients, waves, discontinuities, and boundary layers can be represented.
Spatial and temporal discretization
Spatial discretization divides a domain into nodes, elements, cells, or control volumes. Temporal discretization divides time into steps. The choice of element type, interpolation order, grid spacing, time step, and integration scheme affects accuracy and stability. Fine discretization can reduce truncation error but increase computational cost and sometimes amplify conditioning problems.
Error and convergence
Discretization introduces approximation error. Mesh convergence or time-step convergence studies test whether results become stable as the discretization is refined. A simulation that has not been checked for discretization sensitivity can produce precise-looking but unreliable results. Local refinement may be needed near stress concentrations, shocks, contact regions, boundary layers, or steep gradients.
Signals and sampling
In sampled systems, discretization in time is governed by sampling rate and anti-alias filtering. Quantization then maps amplitude to discrete levels. Both choices affect measurement fidelity, control performance, and spectral analysis. Sampling too slowly can create aliasing that cannot be corrected later.
Common mistakes
Common mistakes include assuming that a finer mesh is automatically correct, refining only globally instead of where gradients matter, using an unstable time step, and ignoring boundary-condition sensitivity. Another error is comparing two simulations with different discretizations but not checking whether observed differences come from physics, numerics, or solver tolerances.