$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
The nonlinear \( 2\times 2 \) system
Introduce \( u_0 \) and \( u_1 \) for \( u_0^{n+1} \) and
\( u_1^{n+1} \), write \( u_0^{(1)} \) and
\( u_1^{(1)} \) for \( u_0^n \) and \( u_1^n \), and rearrange:
$$
\begin{align*}
F_0(u_0,u_1) &=
{\color{red}u_0}
- u_0^{(1)} + \Delta t\,\sin\left(\frac{1}{2}({\color{red}u_1}
+ u_1^{(1)})\right)
+ \frac{1}{4}\Delta t\beta ({\color{red}u_0} + u_0^{(1)})
|{\color{red}u_0} + u_0^{(1)}| =0
\\
F_1(u_0,u_1) &=
{\color{red}u_1} - u_1^{(1)} -\frac{1}{2}\Delta t({\color{red}u_0}
+ u_0^{(1)}) =0
\end{align*}
$$
