Fine and coarse propagators on a common macro grid

Let $d,N\ge 1$, let $X={\mathbb{R}}^d$, and fix macro times $t_0<t_1<\cdots <t_N$. A fine/coarse propagator pair is a pair of deterministic maps $F_n:D_n^F\to X$ and $G_n:D_n^G\to X$, for each $0\le n<N$, where $D_n^F,D_n^G\subseteq X$ are their admitted input sets. Both maps advance a state across the same interval $[t_n,t_{n+1}]$. The labels fine and coarse identify the two roles chosen by the user; they assert no accuracy, stability, cost, or convergence property.

For an initial state $x_0\in X$, whenever every call succeeds, the sequential fine trajectory is the finite sequence $x_{n+1}=F_n(x_n)$, and the sequential coarse trajectory is defined by replacing each $F_n$ with $G_n$. A partially defined map is unavailable outside its admitted set; a failed stage evaluation or an unavailable implicit branch is a propagation failure, not a numerical value.

A canonical ODE instantiation uses the explicit Runge–Kutta stage map of Explicit Runge–Kutta tableaus, stage maps, and failure status for $r\ge 1$ fine substeps of size $(t_{n+1}-t_n)\mathrm{/}r$, composed in order to form $F_n$. A coarse backward Euler value is an endpoint $z$ satisfying the equation in Backward Euler step equation and algebraic residual over the full macro interval. Its branch selector is part of the input and must select exactly one admissible root for each admitted start state; if no such selected root exists, the coarse map is unavailable. The backward Euler definition alone asserts neither root existence nor uniqueness.

This definition distinguishes the sequential fine trajectory from samples of an exact ODE solution. Any estimate of the fine method's error relative to that exact solution is a separate obligation.