For $\delta >0$, we define the filtered space $$\label{FiteredSpaceTime}
        \mathfrak{C}(\delta) = \hbox{span} \{\Phi_j  \hbox{ corresponding to eigenvalues } |\mu_j|\leq \delta \}.$$ One easily checks that this space is left invariant by the equation , and thus we will identify the space of trajectories $y$ solutions of lying in $\mathfrak{C} (\delta)$ with the space of initial data $y^0$ lying in $\mathfrak{C}(\delta)$.

Let us now describe the time-discretization schemes under consideration.

We assume that the time-discretization of system discretized with a time-step $\tau >0$ takes the form $$\label{AbstractTimeDisc}
        y^{k+1}_\tau = {\mathbb{T}}_\tau y^k_\tau, \quad k \in {\mathbb{Z}}, \qquad y^0_\tau = y^0.$$ Here, $y^k_\tau$ denotes an approximation of the solution $y$ of at time $k \tau$, and ${\mathbb{T}}_\tau$ is assumed to be an approximation of $\exp(\tau {{\mathcal{A}}})$.

To be more precise, we assume that ${\mathbb{T}}_\tau$ is a linear operator which has the same eigenvectors as the operator ${{\mathcal{A}}}$, and such that, for some function $f$, $$\label{TonEig}
    {\mathbb{T}}_\tau \Phi_j = \exp (i \mu_{j, \tau} \tau) \Phi_j, \quad \hbox{where } \mu_{j, \tau} = \frac{1}{\tau} f (\mu_j \tau).$$ Actually, this operator may also be written as $${\mathbb{T}}_\tau = \exp\left( i f(- i {{\mathcal{A}}}\tau)\right).$$ Let us now make precise the assumptions we impose on $f$.

First, we assume 