Embedded Runge–Kutta pair and difference estimator

For an ODE $y^{\mathrm{\prime }}=f(t,y)$ on states in ${\mathbb{R}}^d$ with $d\ge 1$, two explicit Runge–Kutta weight vectors $b=(b_i{)}_{i=1}^s$ and $\widehat{b}=({\widehat{b}}_i{)}_{i=1}^s$ are an embedded pair when they use the same explicit stage recursion $K_i=f\text{\hspace{0.17em}⁣}(t_n+c_{i}h,y_n+h{\sum }_{j<i}{a}_{ij}{K}_j),1\le i\le s,$ but produce two one-step values $Y=y_n+h{\sum }_{i=1}^s{b}_i{K}_i,\widehat{Y}=y_n+h{\sum }_{i=1}^s{\widehat{b}}_i{K}_i\mathrm{.}$ The two methods are the high and low members when their local orders are respectively at least $p+1$ and $p$, for an integer $p\ge 1$. Their embedded difference estimator is the nonnegative scalar $E_n(h):=\mathrm{\parallel }\widehat{Y}-Y{\mathrm{\parallel }}_2,$ with the Euclidean norm. This definition asserts only how the estimator is computed; its reliability as a bound for either local defect is a separate claim and is not part of the definition.

The stage recursion and the local defect sign convention are those of Explicit Runge–Kutta tableaus, stage maps, and failure status and One-step ODE methods, local defect, uniform order, and step stability.