The three-stage third-order SSP Runge–Kutta step

Input an initial pair $(t,y)$, a step $h>0$, an open time-state domain $D$, and a right-hand-side evaluator for $f:D\to {\mathbb{R}}^d$. Before each query, check that its time-state pair is in $D$; return domain-failure with that stage index if it is not. If the evaluator reports failure, return evaluation-failure with that stage index. Execute:

1. Query $f(t,y)$. On success set $Y_1=y+hf(t,y)$.
2. Query $f(t+h,Y_1)$. On success set $Y_2=\frac{3}{4}y+\frac{1}{4}(Y_1+hf(t+h,Y_1))$.
3. Query $f(t+h\mathrm{/}2,Y_2)$. On success set $y^{+}=\frac{1}{3}y+\frac{2}{3}(Y_2+hf(t+h\mathrm{/}2,Y_2))$.
4. If $(t+h,y^{+})\mathrm{\notin }D$, return output-domain-failure; otherwise return success with $(t+h,y^{+})$.

The procedure uses only explicit function evaluations; it has no implicit stage solve. If a numerical evaluator can fail even at a domain point, its failure is returned unchanged with the corresponding stage index. The exact mathematical method assumes all queried values exist.