discretisierte

Discretisierte is the German adjective describing something that has been transformed from a continuous form into a discrete representation, typically to enable analysis, simulation, or computation. It is widely used in mathematics, numerical analysis, physics, engineering, and computer science to refer to models, equations, or signals that have been converted from a continuous to a discrete form.

Discretization replaces continuous domains or operators with discrete counterparts. In space, a smooth region is replaced by a grid or mesh of nodes and elements; in time, continuous evolution is approximated by a sequence of time steps. This enables the numerical solution of problems such as differential equations. Common discretization methods include finite difference methods (FDM) on regular grids, finite element methods (FEM) and finite volume methods (FVM) on irregular meshes, and spectral methods that approximate solutions with global basis functions.

Discretized models introduce discretization error, which depends on the mesh size, time step, and the chosen method. Key properties include consistency (the discrete model approximates the original problem as the discretization becomes finer), stability (errors do not grow uncontrollably), and convergence (the solution approaches the true solution as discretization parameters improve). Practical concerns include mesh quality, boundary condition handling, and computational cost. The Courant–Friedrichs–Lewy condition is a notable stability criterion for certain time-dependent schemes.

In German-language contexts, terms like diskretisierte Modelle or diskretisierte Raum-Zeit are used to describe space-time discretization on a lattice, as seen in areas such as computational fluid dynamics or lattice gauge theory. Discretization is distinct from sampling or quantization, which pertain to signal processing, whereas discretization focuses on replacing continuous models or equations with discrete representations suitable for computation.