Chebyshev stage perturbations have an explicit quadratic bound

Let $d\ge 1$, let $A$ be a self-adjoint endomorphism of ${\mathbb{R}}^d$ with its Euclidean inner product and $⟨Av,v⟩\le 0$. For first-order RKC assume the integer stage count $s\ge 1$ and $h\rho \le 2s^2$; for linear second-order RKC assume $s\ge 2$ and $h\rho \le 2(s^2-1)\mathrm{/}3$, where $\rho ={\max }_i(-{\lambda }_i(A))$. In either branch, let $X=I+hA\mathrm{/}{s}^2$ or $X=I+w_{1}hA$, respectively.

Write $Y_j$ for the generic recurrence stage; in the second-order branch this is the $V_j$ of Second-order linear stabilized Runge–Kutta–Chebyshev step. Model the error injected into each recurrence update by vectors ${\delta }_1,\dots ,{\delta }_s$: ${\widehat{Y}}_0=y,{\widehat{Y}}_1=X{\widehat{Y}}_0+{\delta }_1,{\widehat{Y}}_j=2X{\widehat{Y}}_{j-1}-{\widehat{Y}}_{j-2}+{\delta }_j(2\le j\le s)\mathrm{.}$ If $\mathrm{\parallel }{\delta }_j{\mathrm{\parallel }}_2\le \epsilon$ for every $j$, then the final stage error relative to the unperturbed recurrence is at most $\mathrm{\parallel }{\widehat{Y}}_s-Y_s{\mathrm{\parallel }}_2\le \frac{s(s+1)}{2}\epsilon \mathrm{.}$ For linear second-order RKC the returned state multiplies this bound by $b_s\le 1$. An additional output perturbation of norm ${\epsilon }_{\mathrm{o}\mathrm{u}\mathrm{t}}$ adds at most ${\epsilon }_{\mathrm{o}\mathrm{u}\mathrm{t}}$. This is a bound for the stated additive-error model; it does not assert a floating-point error size.

The bound grows quadratically with the number of stages. It applies to self-adjoint operators whose scaled eigenvalues stay in the certified interval. It does not control nonnormal transient growth or replace a separate bound on each arithmetic or matrix-action perturbation.