\left[\begin{array}{c} U_{j+1}^{n,1}\\  U_{j+1}^{n,2} \end{array} \right]=\\
&\left[\begin{array}{c} U^n\\  U^n \end{array} \right]-k
\left[\begin{array}{cc}
a_{11} &  a_{12}\\
a_{21} &  a_{22}
\end{array}\right]\, \left[\begin{array}{c} B(U_{j}^{n,1})\\ B( U_{j}^{n,2}) \end{array} \right]\\
&+k\left[\begin{array}{cc}
a_{11} &  a_{12}\\
a_{21} &  a_{22}
\end{array}\right]\,\left[\begin{array}{cc}
J_1(U^{\ast}) -J(U_j^{n,1}) & 0\\
0 & J_1(U^{\ast}) - J(U_j^{n,2})
\end{array}\right]\,  \left[\begin{array}{c} U_{j+1}^{n,1}\\  U_{j+1}^{n,2} \end{array} \right],
\end{split}$$ for $j\ge 0$, a form that immediately suggests an iterative scheme for approximating $U_{j+1}^{n,i}$, $i=1,2$. This scheme generates [*inner*]{} iterates denoted by $U^{n,i,\ell}_{j+1}$ for given $n,i,j$, and $\ell=0,1,2,\ldots$ ($U^{n,i,\ell}_{j+1}$ approximates $U^{n,i}_{j+1}$) that are found recursively from the equations $$\label{E37}
\left[\begin{array}{cc}
I+k a_{11} J_1(U^{\ast}) & k a_{12}J_1(U^{\ast})\\
k a_{21}J_1(U^{\ast}) & I+k a_{22} J_1(U^{\ast})
\end{array}\right]\, \left[\begin{array}{c} U_{j+1}^{n,1,\ell+1}\\  U_{j+1}^{n,2,\ell+1} \end{array} \right]=\\
\left[\begin{array}{c} r_{j+1}^{n,1,\ell}\\  r_{j+1}^{n,2,\ell} \end{array} \right],$$ for $\ell\ge 0$, where $$r_{j+1}^{n,i,\ell}=U^n-k\sum_{m=1}^2a_{im}B(U^{n,m}_j)+k\sum_{m=1}^2a_{im}(J_1(U^{\ast})-J(U^{n,m}_j)) U_{j+1}^{n,m,\ell}.$$ The linear system (\[E37\]) can be solved efficiently as follows: Since $a_{12}a_{21}<0$, it is possible, upon scaling the matrix on the left-hand side of the system by a diagonal similarity transformation, to write it as $$\label{E38}
\left[\begin{array}{cc}
I+\frac{1}{4} k J_1(U^{\ast}) & k J_1(U^{\ast})/4\sqrt{3}\\
k J_1(U^{\ast})/4\sqrt{3} & I+\frac{1}{4}k  J_1(U^{\ast})
\end{array}\right]\, \left[\begin{array}{c} U_{j+1}^{n,1,\ell+1}\\ \mu  U_{j+1}^{n,2,\ell+1} \end{array} \right]=\\
\left[\begin{array}{c} r_{j+1}^{n,1,\ell}\\  \mu r_{j+1}^{n,2,\ell} \end{array} \right],$$ where $\mu=2-\sqrt{3}$. The system (\[E38\]) is equivalent to the single complex $N\times N$ system $$\label{E39}
(I+k\zeta  J_1(U^{\ast}))Z=R,$$ where $\zeta=\frac{1}{4}+{\rm i}/4\sqrt{3}$, and where 