Diskretisering

Diskretisering, or discretization, is the process of transforming a continuous mathematical model into a discrete one suitable for computation. It replaces continuous domains, variables, and operators with a finite set of points and algebraic relations. Discretization can be applied in time, space, or both, and it is a central step in numerical methods for solving differential equations and dynamic systems.

Spatial discretization replaces a continuum with a mesh or grid. Common methods include finite difference (regular grids), finite element (unstructured meshes with variational formulation), finite volume (conserving fluxes across control volumes), and spectral methods (global basis functions). Temporal discretization replaces continuous time with a sequence of time steps, using explicit or implicit schemes such as Euler, Runge–Kutta, or backward differentiation formulas. Often both dimensions are discretized, yielding a system of algebraic equations.

Key concerns are discretization error, stability, and convergence. Truncation error arises from approximating derivatives or integrals. Stability conditions (for example CFL conditions in hyperbolic problems) determine whether errors grow or decay. Consistency and convergence describe how the discrete solution approaches the continuous solution as the mesh size or time step decreases. Applications span fluid dynamics, heat conduction, structural mechanics, electromagnetism, and image processing. Choosing an appropriate discretization involves trade-offs between accuracy, computational cost, and problem features such as irregular geometry or sharp gradients. Adaptive mesh refinement and error estimation are common tools to improve efficiency.