timestep

A timestep, often denoted Δt, is a discrete increment of time used to advance the state of a system in a simulation or numerical calculation. It defines how much simulated time passes between successive updates. In many contexts, Δt is expressed in seconds and sets the pace of the simulation.

In numerical integration of ordinary differential equations and dynamic systems, the timestep discretizes time. Common schemes include Euler, Runge-Kutta, and Verlet. Fixed timesteps use a constant Δt, while adaptive timestepping adjusts Δt based on error estimates to balance accuracy and performance. The choice of timestep affects stability and accuracy, with smaller steps generally improving precision but increasing computational cost. For stiff or highly nonlinear problems, stability constraints can require very small Δt or specialized methods.

The timestep also appears in stochastic and discrete-time models, where state changes occur in steps of Δt. The choice of Δt affects the accuracy of statistics such as mean and variance, and in continuous-time models it acts as a numerical proxy for real time. In simulations of random processes, timestep size can influence convergence to theoretical results.

In real-time simulation, video games, and animation, a fixed timestep improves determinism across platforms; a common pattern uses an accumulator to handle variable frame rates while updating physics in fixed increments. If the timestep is too large, simulations can become unstable or physically implausible; if too small, performance suffers.

In practice, Δt is constrained by stability conditions (for example the CFL condition in certain PDEs) and by computational limits. In fields such as weather and climate modeling, very small timesteps or multi-rate schemes are used to resolve fast processes alongside larger-scale dynamics.