Embedded pairs and adaptive stepping

An embedded pair computes two endpoint approximations from one collection of stage derivatives. If the lower formula has order p minus one and the higher has order p, their exact-start defects typically begin at powers h to the p and h to the p plus one. Their difference is therefore dominated by the lower formula's defect, provided its leading coefficient does not vanish on the current problem. Propagating the higher result does not change which defect dominates that difference.

This distinction prevents an incorrect controller exponent. A difference behaving like h squared calls for a square-root step adjustment, even if the accepted endpoint uses a second-order formula with a cubic local defect. Some solvers use special estimators with different powers, so the exponent should come from the actual estimator rather than the displayed method name. Accidental cancellation, nonsmooth forcing, and unresolved fast scales can make the asymptotic model unreliable on a particular attempted step.

A raw error norm can be meaningless when the state mixes quantities with very different units and magnitudes. Give each component an absolute tolerance with its physical units, plus a dimensionless relative tolerance. For this lesson's illustrative controller, scale by the absolute tolerance plus the relative tolerance times the larger old or proposed component magnitude. Dividing the estimated error by that scale produces a dimensionless number. Real libraries may choose a different comparable scaling convention or a weighted root-mean-square norm.

Absolute tolerance is essential near zero, where a purely relative test would demand unrealistic precision or divide by zero. Conversely an absolute tolerance larger than the variation of a small but important component can allow that component to be poorly resolved. Choose scales from the scientific question rather than from whatever values happen to appear in the first run. An acceptance test bounds a normalized estimate, not every true component error or every accumulated error in a derived observable.

Assuming the error scale changes slowly and the estimate follows a power law, multiply the step by the inverse qth power of the normalized error. A safety factor below one makes the proposal conservative, and growth or shrinkage limits prevent extreme jumps. A failed step is rejected: retain the previous accepted state and retry with a shorter step. Accepting the inaccurate trial and merely choosing a shorter next step would leave the already introduced error in the trajectory.

At zero estimated error a controller uses a capped growth rule rather than dividing by zero. Repeated rejections can indicate stiffness, bad scaling, a discontinuity, or a failing nonlinear solve rather than a need for indefinitely smaller steps. Practical controllers also smooth successive estimates to reduce oscillation in step size. Final-time clipping, minimum representable increments, and event handling impose further constraints. These implementation details explain why an adaptive solver is more than a pair of coefficient vectors.