First-order stabilized explicit Runge–Kutta–Chebyshev step

This algorithm takes an autonomous vector field $f$ on an open state domain $V\subseteq {\mathbb{R}}^d$, an admitted initial state $y$, a step $h>0$, an integer stage count $s\ge 1$, and an evaluator that returns either $f(u)$ or an evaluation-failure status. In the exact mathematical model the evaluator returns $f(u)$ at every $u\in V$; the failure status records only an unavailable numerical evaluation. Define stage abscissae $c_j=j^2\mathrm{/}{s}^2$ for $0\le j\le s$. Set $Y_0=y$. Before each query check that its stage state belongs to $V$; if it does not, return domain-failure with that stage index. If the evaluator reports failure, return evaluation-failure with that stage index. Query at $Y_0$ and, on success, set $Y_1=Y_0+(h\mathrm{/}{s}^2)f(Y_0)$. For $j=2,\dots ,s$, query at $Y_{j-1}$ and, on success, set $Y_j=2Y_{j-1}-Y_{j-2}+\frac{2h}{s^2}f(Y_{j-1})\mathrm{.}$ If the final state $Y_s$ is outside $V$, return output-domain-failure; otherwise return success with state $Y_s$.

For a nonautonomous field $f:D\to {\mathbb{R}}^d$ on an open time-state domain $D\subseteq \mathbb{R}\times {\mathbb{R}}^d$, input an admitted pair $(t,y)\in D$ and an evaluator that returns either $f(\tau ,u)$ or evaluation-failure. In the exact mathematical model it returns $f$ at every point of $D$. Use the same state recurrence, querying at the declared stage times: query at $(t+c_{0}h,Y_0)=(t,y)$ before setting $Y_1=Y_0+(h\mathrm{/}{s}^2)f(t,Y_0)$; for $j=2,\dots ,s$, query at $(t+c_{j-1}h,Y_{j-1})$ before setting $Y_j=2Y_{j-1}-Y_{j-2}+\frac{2h}{s^2}f(t+c_{j-1}h,Y_{j-1})\mathrm{.}$ Before each query, check that its time-state input lies in $D$; return domain-failure with that stage index if it does not, or evaluation-failure with that index if the evaluator fails. After the stages, return output-domain-failure if $(t+h,Y_s)\mathrm{\notin }D$; otherwise return success with $(t+h,Y_s)$. The scalar linear stability polynomial is established separately.

The stability analysis uses $T_s$ and $U_s$ from Chebyshev polynomials of the first and second kinds by their three-term recurrences; this item defines the executable recurrence and does not itself assert an order or stability theorem.