timestepped

Timestepped refers to models, simulations, or processes in which time is represented as a sequence of discrete increments rather than as a continuous variable. In a timestepped approach, state variables are updated at regular or adaptive time steps by applying a time-stepping scheme that approximates the evolution of the system according to its governing equations, typically ordinary differential equations or differential-algebraic equations.

Common timestepping schemes include explicit methods (for example, forward Euler and Runge-Kutta of various orders), implicit methods (such as backward Euler and implicit Runge-Kutta), and multi-step methods (such as Adams-Bashforth and Adams-Moulton). The choice affects stability, accuracy, and stiffness handling. Step size selection may be fixed or adaptive, guided by error estimation and stability constraints such as the CFL condition in partial differential equations.

Timestepped models are used across computational physics, engineering, climate and weather modeling, structural dynamics, electrical circuit simulation, and computer graphics or game physics where continuous-time processes are approximated by discrete steps. In signal processing, discrete-time models are naturally timestepped with a sampling rate.

Advantages of timestepping include simplicity and compatibility with digital computation, straightforward implementation, and explicit control over error through step size. Disadvantages can include accumulation of numerical error, potential stability issues for stiff or highly dynamic systems, and increased computational cost for small time steps or when fine resolution is required.

Terminology often contrasts timestepped (discrete-time) models with continuous-time formulations. Implementations typically require careful selection of the stepping scheme, step size strategy, and, for coupled or multidimensional problems, consistent data exchange between components.