Adaptive Step Size

Differential Equations · Axiom Academy

A spacecraft on an eccentric orbit swings close past Earth, then drifts far out and back — one step size everywhere is either too slow or too sloppy. Put the fix in your hands. Your spacecraft is riding an elliptical orbit: a close swing past Earth at 0.3 orbit-radii , then out to 1.7 orbit-radii and back, every one period — three moves with the same idea: feel it , solve it , weigh it . One orbit, two very different neighborhoods Near Earth the pull is fierce and the path curves fast. Far out, gravity is weak and the path is almost straight. The equation of motion is the same the whole way around: Pick a FIXED step count and fly the orbit once — watch the markers land: clumped and useless out at apoapsis, too sparse to trust near the close approach. The rule that decides the step Every step, the solver takes ONE step of size h and estimates its own error by comparing it against two half-steps of size h/2 . If the error is too big for your tolerance, shrink h and retry; if it's comfortably small, grow h for next time. Same accuracy, a fraction of the steps Hold the final-position accuracy fixed and ask: how many steps did each approach need to get there? Drag the accuracy target and compare. One rule, everywhere a solver runs: estimate the error, then shrink or grow the step to match it. Small steps buy accuracy exactly where the solution is changing fast; large steps buy speed everywhere else — for free, automatically, without you ever choosing a single fixed step size.

This is the written version of the interactive lesson above. See the full Differential Equations course.