Dense output and event location

A dense-output formula usually combines stage information into a polynomial or another continuous extension over one accepted step. Its degree and accuracy are method-specific and need not equal the endpoint order. Merely joining accepted endpoints with straight lines can lose substantial information: even exact endpoint values do not determine the curve between them. For the exact solution y=t squared on the unit interval, the line through its endpoints gives one half at the midpoint instead of one quarter.

Requested output times describe where values should be returned, not necessarily the internal step schedule. A solver can take a longer accepted step and evaluate its dense extension at several requested times inside it. Increasing the number of saved points therefore need not improve the underlying integration. To improve accuracy, control the integration and interpolation tolerances or the maximum step, then verify the quantity of interest. A smooth-looking densely sampled plot is not by itself evidence that small-scale dynamics were resolved.

Let an event be a zero of $g(t,y(t))$. At a simple crossing, the derivative along the solution is $g_t+g_y f$ and is nonzero. A small perturbation in the event function then shifts the root by approximately the perturbation divided by this crossing speed. A nearly tangent crossing is poorly conditioned: a small state error may produce a large time error or eliminate the crossing. A tight scalar root tolerance cannot fix that sensitivity.

Separate the error in the dense state from the residual left by the scalar root finder. If the true event function is monotone with derivative magnitude at least gamma on an interval containing both roots, a uniform event-function approximation error epsilon bounds the root shift by epsilon divided by gamma, with a scalar root residual added to epsilon when needed. This argument requires a nearby isolated root and the derivative bound; it does not apply unchanged to multiple roots or arbitrary discontinuous event functions.

Many solvers detect a candidate event by a sign change across an accepted step, then refine its time using dense output. Two crossings within one step can leave equal signs at the endpoints and be missed. A tangential root can touch zero without changing sign at all. A direction filter selects upward or downward crossings once found; it does not prove that no unobserved crossing occurred. Known bounds on event frequency or a suitable maximum step can help, but the model must supply that information.

An event that changes the state or the differential equation creates a new initial-value problem. Locate the event, apply the specified jump or mode switch, and restart with consistent state and fresh solver history. Continuing an old smooth interpolant through a discontinuous reset gives a curve for the wrong dynamics. Likewise a zero at the initial time may require a convention to avoid repeatedly rediscovering the same event. Event semantics belong to the modeled system as well as to the root-finding algorithm.