$\Theta_2: S_h\rightarrow S_h$ that satisfy for $v,w\in S_h$ and for all $\chi \in S_h$ $$(B(v,w),\chi)=\frac{1}{2}(\beta vw,\chi') =-\frac{1}{2}(\beta(vw)_x,\chi),$$ $$(\Theta_1 v,\chi)=(\alpha v-\delta v_{xx},\chi'),$$ and $$(\Theta_2 v,\chi)=-(\gamma P_N \mathcal{G} v_x,\chi').$$ If we put $$F(v,w):=B(v,w)+\Theta_1 v+ \Theta_2 v,$$ we see that $$F(v):=F(v,v)=B(v)+\Theta_1 v+\Theta_2 v,$$ where $B(v)=B(v,v)$. The initial-value problem (\[E31\]) is stiff. It is discretized in the temporal variable by the 2-stage Gauss-Legendre implicit Runge-Kutta method, which is fourth-order accurate and has good nonlinear stability properties. It corresponds to the Butcher table $$\begin{array}{cc|c}
a_{11} & a_{12} & \tau_1\\
a_{21} & a_{22} & \tau_2 \\
\hline
b_1 & b_2 & \\
\end{array}\,\,\,\, =\,\,\,\,
\begin{array}{cc|c} \frac{1}{4} & \frac{1}{4}-\frac{1}{2\sqrt{3}} &
\frac{1}{2}-
\frac{1}{2\sqrt{3}} \\
\frac{1}{4}+ \frac{1}{2\sqrt{3}} & \frac{1}{4} & \frac{1}{2}+
\frac{1}{2\sqrt{3}}\\
\hline
\frac{1}{2} & \frac{1}{2} & \\
\end{array}.$$ The fully discrete scheme is now specified more precisely. Let $t^n=nk$, $n=0,1,\ldots, M$, where $T=Mk$. We seek $U^n$ approximating $u_{h}(t^{n})$, and $U^{n,i}$ in $S_h$, $i=1,2$, as solutions of the system of nonlinear equations $$\label{E32}
U^{n,i}=U^n+k\sum_{j=1}^2 a_{ij} F(U^{n,j}),\quad i=1,2,\quad 0\leq n\leq M-1,$$ and set $$\label{E33}
U^{n+1}=U^n+k\sum_{j=1}^2 b_j F(U^{n,j}), \quad 0\leq n\leq M-1,$$ where $U^{0}=u_{h}(0)$. At each time step we solve the nonlinear system (\[E32\]) using Newton’s method as follows. Given $n \ge 0$, let $U_0^{n,i}\in S_h$, $i= 1, 2$ be an 