Dormand-Prince method — error estimation and tolerance-driven adaptation
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Adaptive step size control uses two RK methods of different orders to estimate the local error.
The Dormand-Prince RK45 computes a 4th and 5th order solution simultaneously (6 function evaluations + FSAL).
The error estimate ε drives the step size update: h_new = h·(tol/ε)1/5·safety.
Small tolerance → many small steps → high accuracy; large tolerance → fewer, larger steps but less accurate.
Stiff equations force tiny steps even when the solution is smooth — explicit methods are inefficient there.