timestepping

Timestepping refers to the numerical procedure used to advance the solution of time-dependent equations in discrete time increments. It is a key component in solving initial-value problems for ordinary differential equations (ODEs) and, after spatial discretization, in solving time-dependent partial differential equations (PDEs). The goal is to approximate the state of a system at times t0, t1, t2, ... by applying a time-stepping scheme that uses information from previous states.

Most timestepping methods fall into explicit or implicit families. Explicit methods compute the new state directly from known quantities at the current or previous steps; examples include Euler's method and explicit Runge-Kutta schemes such as RK4. Implicit methods involve solving an equation for the new state, often requiring iterative or linear solves; examples include backward Euler and Crank–Nicolson. Multistep methods, such as Adams-Bashforth (explicit) and Adams-Moulton (implicit), reuse several past states to achieve higher order with potentially larger time steps. Backward differentiation formulas (BDF) are another common implicit family, particularly for stiff problems.

Stability, accuracy, and efficiency govern the choice of timestepping method. The order of accuracy determines how the error decreases as the time step shrinks. Stability considerations, including stability regions and the Courant–Friedrichs–Lewy (CFL) condition, often constrain the allowable time step, especially for explicit schemes or stiff systems. Adaptive time stepping uses error estimation to adjust the step size dynamically, balancing computational cost against the desired accuracy.

Applications span fluid dynamics, structural dynamics, chemical kinetics, and many fields requiring time-resolved simulations. In some contexts, structure-preserving or symplectic schemes are preferred to conserve invariants over long simulations.