Adaptive time stepping and event detection

Let a method of global order p start from the exact same state and advance through an interval h. One coarse step and two half steps share the leading solution but have different leading errors. Under a smooth asymptotic error expansion, their difference divided by two to power p minus one estimates the error remaining in the finer result, up to sign convention. This is an estimate for one interval, not a certified global bound accumulated over an entire simulation.

Here local state error means the state difference after one step begun from exact input; it is typically order p plus one. Some texts divide that defect by h when defining local truncation error, giving order p instead. A step controller often proposes an increment based on the local state-error exponent and adds a safety factor and growth limits. Near discontinuities or singular behavior, the smooth error expansion can fail and an apparently small estimate may be misleading.

Practical solvers weight each component's error by an absolute tolerance plus a relative tolerance times a characteristic component magnitude. Absolute tolerances carry the component's units; positions and velocities usually need different weights. Accepting each step against that norm does not assert that the final global error equals the tolerance. Errors propagate through the differential equation, and unstable or chaotic sensitivity can amplify earlier defects even when local estimates were reasonable.

For a rapidly decaying component, an explicit integrator can be limited by stability rather than by the desired accuracy of the slow observed motion. Explicit Euler for decay at rate one thousand per second requires a step no larger than two milliseconds for the usual absolute-stability interval; the endpoint is marginal and does not reproduce physical decay. An implicit method may remove that particular stability barrier, yet still needs sufficiently small steps to resolve transients or events. Stability alone is not accuracy.

An event function can represent height above ground, a separation threshold or a turning-point condition. Sign changes between accepted endpoints bracket an odd number of crossings for a continuous function, but several crossings in one step can cancel in sign. A grazing event may touch zero without changing sign at all. Event direction, maximum step, dense-output interpolation and the physical stopping rule therefore belong in the numerical specification.

Root refinement usually acts on an interpolant of the numerical trajectory. Once that root is tightly located, remaining timing error can still come from trajectory and interpolation error; a shallow event derivative makes timing especially sensitive. The root-finding lab demonstrates bracket refinement for one monotone quadratic only. It has no adaptive differential-equation trajectory, dense output or grazing-event search. A fresh Reset and both parameter settings are required because changing its controls retains the elapsed lab clock.