of the solution $u$ of (\[E21\]) is a real $S_{h}$-valued function $u_{h}(t)$ of $t\geq 0$ defined by the ode initial-value problem $$\label{E23}
\begin{split}
&({u_h}_t,\chi)+(\alpha {u_h}_x+\beta u_h {u_h}_x,\chi)+\gamma(P_N\mathcal{G}{u_h}_x,\chi_x)+\delta({u_h}_{xx},\chi_x)=0,\forall \chi\in S_{h}, t\geq 0,\\
&u_h(0)=P_h u_{0},
\end{split}$$ where $P_h$, $P_N$ are the $L^2$ projections onto $S_h$ and $S_N$, respectively, given for $w\in L^{2}$ as $$(P_h w,\chi)=(w,\chi),\quad \forall \chi\in S_h$$ and $$(P_N w,\phi)=(w,\phi),\quad \forall \phi\in S_N,$$ where $(\cdot,\cdot)$ is the $L^{2}(0,2\pi)$ inner product. For $f\in L^{2}$, $P_{N}f$ is represented by $$P_N f(x)=\sum_{k=-N/2}^{N/2-1} \hat{f}_k e^{{\rm i}kx},$$ where $\hat{f}_{k}=\frac{1}{2\pi}\int_{0}^{2\pi} f(x)e^{-ikx}dx, k\in \mathbb{Z}$ are the Fourier coefficients of $f$. Note that $\widehat{(\mathcal{G}f)}_{k}=-{\rm i}{\mathop{\operator@font sign}}(k)\hat{f}_{k}$ and that $\mathcal{G}$ is antisymmetric in $L^{2}$.

The fully discrete hybrid scheme {#sec22}
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We define our fully discrete hybrid scheme following the derivation of the analogous scheme of [@BDKMc] in the case of the generalized KdV equation. (This scheme was also used in [@BDM].) Denoting again by $(\cdot,\cdot)$ the $L^{2}(0,2\pi)$ inner product, we define, for each $t\in [0,T]$, the map $F:S_{h}\rightarrow S_{h}$ by the equation $$(F(u_h),\chi)=-[(\alpha {u_h}_x+\beta u_h {u_h}_x,\chi)+\gamma(P_N\mathcal{G}{u_h}_x,\chi_x)+\delta({u_h}_{xx},\chi_x)], \quad \forall \chi\in S_{h}.$$ Then, the initial-value problem (\[E23\]) may be written as $$\begin{aligned}
{u_h}_t=F(u_h),\quad 0\leq t\leq T,\quad u_{h}(0)=P_{h}u_{0}.\label{E31}\end{aligned}$$ In addition to $F$ we define the maps $B:S_h\times S_h\rightarrow S_h$, $\Theta_1:S_h\rightarrow S_h$ and 