tidssteget

Tidssteget, the Swedish term for time step, denotes the discrete increment of time used to advance the state of a dynamical model in numerical simulations. In time-dependent problems, the solution is computed at a sequence of times t0, t1 = t0 + Δt, t2 = t1 + Δt, and so on. The choice of Δt influences accuracy, stability, and computational cost.

In explicit time-stepping schemes such as forward Euler or classical Runge–Kutta, Δt is often limited by stability constraints, for example the Courant–Friedrichs–Lewy (CFL) condition in partial differential equations. Smaller Δt improves local truncation error and accuracy but increases the number of steps. Implicit schemes, like backward Euler or Crank–Nicolson, can remain stable for stiffer systems and permit larger Δt, at the expense of solving algebraic equations at each step.

Adaptive time stepping dynamically adjusts Δt based on error estimates, balancing precision and efficiency. Time-step control is common in simulations of fluid dynamics, electromagnetism, molecular dynamics, climate models, and financial engineering.

Practical considerations include the desired error tolerance, smoothness of the solution, stiffness of the governing equations, and computational resources. Time-step selection is a central aspect of numerical analysis and is often guided by stability analyses and empirical testing.

See also: time integration, numerical stability, CFL condition, adaptive timestep, explicit method, implicit method, finite difference method.