Explicit Runge–Kutta tableaus, stage maps, and failure status

Let $d\ge 1$, let $D\subseteq \mathbb{R}\times {\mathbb{R}}^d$ be an open time-state domain, and let $f:D\to {\mathbb{R}}^d$ be the right-hand side of a first-order initial-value problem in the sense of First-order systems, initial value problems, and solutions on intervals. An explicit $s$-stage Runge–Kutta tableau consists of a positive integer $s$, real coefficients $a_{ij}$, $b_i$, and $c_i$ with $a_{ij}=0$ whenever $j\ge i$, and $c_i={\sum }_{j<i}{a}_{ij}(1\le i\le s)\mathrm{.}$ The finite row sum is empty, hence $c_1=0$.

For a step size $h>0$ and an admitted base pair $(t,y)\in D$, the stage graph consists of the finite tuples satisfying $Y_i=y+h{\sum }_{j<i}{a}_{ij}{K}_j,t_i=t+c_{i}h,K_i=f(t_i,Y_i)(i=1,\dots ,s)\mathrm{.}$ These equations characterize each stage from the preceding slopes because $a_{ij}=0$ for $j\ge i$. For a complete stage tuple, the candidate endpoint is $y^{+}=y+h{\sum }_{i=1}^s{b}_i{K}_i\mathrm{.}$

To include unavailable evaluations, let $(E_i{)}_{i=1}^s$ be evaluator response functions on $D$: at each point $E_i$ has value either $f(\tau ,u)$ or the symbol evaluation-failure for stage $i$. Stage-dependent responses allow availability to differ between repeated inputs. A successful prefix of length $r$, $0\le r\le s$, is a stage tuple through index $r$ whose stage pairs lie in $D$ and whose responses under $E_i$ equal the corresponding slopes $K_i$. The empty prefix is allowed. For any prefix of length $r<s$, the next pair $(t_{r+1},Y_{r+1})$ is characterized by the same displayed equations using its already specified slopes. There is a unique maximal successful prefix, since every admissible successful extension has the uniquely specified slope $f(t_i,Y_i)$.

The status-labelled stage map is the following mathematical map. For a maximal prefix of length $r<s$, its value is domain-failure with index $r+1$ when the next pair is outside $D$, and evaluation-failure with index $r+1$ when that pair lies in $D$ and its response under $E_{r+1}$ is evaluation-failure. For a complete prefix its value is domain-failure at output when $(t+h,y^{+})\mathrm{\notin }D$, and success with $(t+h,y^{+})$ otherwise. Thus no slope at an inadmissible pair is part of the stage graph. In the exact mathematical model $E_i=f$ on $D$ for every $i$; evaluation-failure represents only an unavailable numerical evaluation. On its success domain the state component $y^{+}$ defines a one-step map in the sense of One-step ODE methods, local defect, uniform order, and step stability.