\tau\right) y^0, \quad \hbox{ where }    f(\alpha)= 2 \arctan\left( \frac{6\alpha}{12-\alpha^2}\right).$$ Hence the discretization fits the assumption of Theorem \[Thm-Main\] by setting $$f : (-2 \sqrt{3}, 2 \sqrt{3}) \to {\mathbb{R}}; \quad f(\alpha)= 2 \arctan\left( \frac{6\alpha}{12-\alpha^2}\right).$$ Note that, here, the function $f$ is limited to the range where $R = 2\sqrt{3}$.

The Newmark method for second order in time equations
-----------------------------------------------------

The Newmark method is designed for second order in time equations such as, in particular, the wave equation. Namely, let ${{\mathcal{A}}}_0$ be a self-adjoint operator defined on an Hilbert space ${{\mathcal{X}}}_0$ with dense domain $\mathcal{D}({{\mathcal{A}}}_0)$ and with compact resolvent, and consider the following equation: $$\label{AbstractWaveCont}
        \varphi'' + {{\mathcal{A}}}_{0} \varphi = 0, \quad t \in {\mathbb{R}}, \qquad (\varphi, \varphi')(0) = (\varphi^0, \varphi^1) \in \mathcal{D}({{\mathcal{A}}}_0^{1/2})\times {{\mathcal{X}}}_0.$$ The Newmark method yields, for $\beta \in [0,1/4]$, the following time-discrete scheme: $$\label{Anewmark}
        \left\{
            \begin{array}{ll}
                {\displaystyle}\frac{\varphi_\tau^{k+1}+\varphi_\tau^{k-1}-2\varphi_\tau^k}{\tau^2}
                +
                {{\mathcal{A}}}_0\left( \beta \varphi_\tau^{k+1}+(1-2\beta )\varphi_\tau^{k}+ \beta \varphi_\tau^{k-1}\right)=0,
            \smallskip
                \\
                {\displaystyle}\left( \frac{\varphi_\tau^0+\varphi_\tau^1}{2}, \frac{\varphi_\tau^1-\varphi_\tau^0}{\tau} \right) = (\varphi^0, \varphi^1)
            \in {{\mathcal{X}}}_0^2.
            \end{array}
        \right.$$ System is conservative and preserves the discrete energy: $$\begin{gathered}
\label{EnergyNewmark}
    E^{k+1/2} =  
        {\left\Vert{{\mathcal{A}}}_0^{1/2} \left(\frac{\varphi_\tau^k + \varphi_\tau^{k+1}}{2} \right)  \right\Vert}_{{{\mathcal{X}}}_0}^2 
        +
        {\left\Vert\frac{\varphi_\tau^{k+1} - \varphi_\tau^k}{\tau} \right\Vert}_{{{\mathcal{X}}}_0}^2 
        \\
        +
        (4 \beta - 1) \frac{\tau^2}{4} {\left\Vert{{\mathcal{A}}}_0^{1/2} \left(\frac{\varphi_\tau^{k+1}- \varphi_\tau^k}{\tau} \right)\right\Vert}_{{{\mathcal{X}}}_0}^2.
    \end{gathered}$$

System fits the abstract setting by 