Second-order linear stabilized Runge–Kutta–Chebyshev step

Input a finite-dimensional real vector space $V$, a linear endomorphism $A:V\to V$, an initial vector $y\in V$, a step $h>0$, and an integer stage count $s\ge 2$. Each application of $A$ is the exact value of this total linear map, so this mathematical procedure has no application-failure branch. Define $w_1=\frac{3}{s^2-1},b_s=\frac{s^2-1}{3s^2},a_s=1-b_s\mathrm{.}$ Set $V_0=y$ and $V_1=V_0+w_{1}hA(V_0)$. For $j=2,\dots ,s$, compute $V_j=2V_{j-1}+2w_{1}hA(V_{j-1})-V_{j-2}\mathrm{.}$ Return $a_s{V}_0+b_s{V}_s$. Each stage uses one application of $A$; no implicit system is solved. The output is the one-step update for the autonomous linear equation $y^{\mathrm{\prime }}=Ay$. Here $T_s$ denotes the Chebyshev polynomial defined by Chebyshev polynomials of the first and second kinds by their three-term recurrences; the equivalent polynomial in $hA$ and its certified real stability interval are proved separately.