Time Integration Methods
Time integration advances a simulation from one instant to the next; the scheme chosen sets its stability, accuracy, and cost.
Marching through time
A time-dependent simulation computes the system’s state at a sequence of instants, each derived from the last. The rule for taking that step is a time integration method, and its properties determine whether the simulation is accurate, stable, and affordable.
Explicit versus implicit
- Explicit: compute the next state directly from the current one — cheap per step but often limited to small stable timesteps.
- Implicit: solve an equation involving the next state — costlier per step but stable at much larger timesteps.
Stiffness
A system is stiff when it contains both very fast and very slow processes. Explicit methods must then take tiny steps to stay stable even if the fast process is not of interest, which makes implicit methods, stable at large steps, the practical choice for stiff problems.
Order of accuracy
Higher-order methods reduce error faster as the timestep shrinks, giving more accuracy per unit of computation — up to a point. The right order balances accuracy against complexity and against the accuracy of the rest of the simulation; a high-order time step paired with a coarse mesh wastes effort.
Choosing well
No single method is best for everything. Matching the integrator to the problem’s stiffness, accuracy needs, and stability constraints is a core modeling decision. Getting it wrong shows up as a simulation that either diverges or crawls, which is why time integration sits at the heart of numerical practice.