Step-doubling error estimation (${\hat{e}}_{\mathrm{fine}}=(Y_{h/2,h/2}-Y_h)/(2^p-1)$)

For an order-$p$ one-step method with leading numerical error $C{h}^{p+1}$, two half steps have leading error $C{h}^{p+1}/2^p$. The displayed difference estimates exact minus the finer approximation. Adding it implements Richardson extrapolation. It requires a common starting value, the smooth local expansion, and accuracy of all algebraic solves; it can fail across a nonsmooth event.