◆ advanced25 mindraft

Letting the solver choose the step

Fixed steps waste effort where the solution is smooth and under-resolve where it is not. Estimate the local error each step and resize.

assumes you already have

Forward Euler, and why it is not enough RK4, and where the magic numbers come from

Planned outline

Estimating local error by taking the same interval two ways.
Embedded pairs: two weight rows over shared stages, so the estimate is nearly free.
Dormand-Prince 5(4) - what ode45 and SciPy RK45 actually run.
The step size controller: PI control on the error, safety factors, rejected steps.
Absolute versus relative tolerance, and why getting this wrong is the most common solver misuse.
Dense output: interpolating within a step without shrinking it.

Interactives planned

Each numbered point above gets at least one thing you can manipulate, not a figure. See AGENTS.md for the widget vocabulary and the rule that every lesson ships at least one graded interaction and one recall card.