Second-order nonlinear stabilized Runge–Kutta–Chebyshev step

Input a first-order right-hand side $f:D\to {\mathbb{R}}^d$ on an open time-state domain $D\subseteq \mathbb{R}\times {\mathbb{R}}^d$, an admitted pair $(t,y)\in D$, a step $h>0$, an integer stage count $s\ge 2$, and a right-hand-side evaluator that returns either $f(\tau ,u)$ or an evaluation-failure status at each queried $(\tau ,u)\in D$. In the exact mathematical model the evaluator returns $f$ at every point of $D$; the failure status records only an unavailable numerical evaluation. Set $w_1=\frac{3}{s^2-1},b_0=b_1=b_2=\frac{1}{4},b_j=\frac{j^2-1}{3j^2}(3\le j\le s),a_j=1-b_j(0\le j\le s)\mathrm{.}$ For the stages define $c_0=0$, $c_1=b_1{w}_1$, and $c_j=b_j{w}_1{j}^2$ for $2\le j\le s$. Put $Y_0=y$. Before each query check that its time-state input belongs to the declared domain; if not, return domain-failure with that stage index. If the right-hand-side evaluator reports failure, return evaluation-failure with that stage index. Query $k_0=f(t,Y_0)$ and on success set $Y_1=Y_0+b_1{w}_{1}h{k}_0$. For each $j=2,\dots ,s$, define ${\mu }_j=\frac{2b_j}{b_{j-1}},{\nu }_j=-\frac{b_j}{b_{j-2}},m_j=\frac{2b_j{w}_1}{b_{j-1}},g_j=-a_{j-1}{m}_j,$ query $k_{j-1}=f(t+c_{j-1}h,Y_{j-1})$, and on success set $Y_j=(1-{\mu }_j-{\nu }_j)y+{\mu }_j{Y}_{j-1}+{\nu }_j{Y}_{j-2}+m_{j}h{k}_{j-1}+g_{j}h{k}_0\mathrm{.}$ If $(t+h,Y_s)$ is outside the declared domain, return output-domain-failure; otherwise return success with $(t+h,Y_s)$. All stages use only the initial state and already computed values, so the procedure contains no implicit solve.

For an autonomous field, omit the time arguments. Order, stability, and stage-amplification properties are established separately.

The coefficient $b_2$ agrees with the displayed formula at $j=2$; it is listed among the base values to make the recurrence unambiguous. The Chebyshev identities used to derive these coefficients are supplied by Chebyshev polynomials of the first and second kinds by their three-term recurrences.