f_h[M^{n+1}]_{i,j} ,\\[6pt]
D_{\Delta t}^\alpha M_{i,j}^{n+1} - (\Delta_h M^{n+1})_{i,j} - {\mathcal{B}}_{i,j}(U^{n},M^{n+1}) = 0 ,\\[6pt]
U^{N_T}_{i,j} = u_T(x_{i,j}), \,   M_{i,j}^{0}=  \frac 1 {h^2} \int_{|x-x_{i,j}|_{\infty}\le h/2} m_0(x) dx,
\end{array}
\right.$$ for $n=0,\dots,N-1$ and $0\le i,j\le N_h$.\
We prove some results for the problems and which will be used in the following.

\[le:existence\] Let $V$ be a grid function on ${\mathbb{T}}_h^2$ and $\delta$ be a positive parameter. Then, there exists a unique grid function $U$ such that $$\label{eq:stationary}
\delta U_{i,j} - (\Delta_hU)_{i,j} + g(x_{i,j},[D_hU]_{i,j}) = V_{i,j}.$$

For the proof of the previous result, we refer to [@acd Lemma 1].

\[prop:ex\_HJ\] There exists a function $U=(U^n)_{n=0,\dots,N}$ which satisfies . Moreover, if $U= (U^n)_{ 0\le n\le N }$, $V= (U^n)_{ 0\le n\le N }$ are such that $U^N\le V^N$ and $$\begin{aligned}
\bar D_{\Delta t}^{\alpha} U_{i,j}^{n} - (\Delta_h U^{n})_{i,j} + g(x_{i,j},[D_hU^{n}]_{i,j}) \le f_h[M^{n+1}]_{i,j},\quad n=0,\dots,N-1,\\
\bar D_{\Delta t}^{\alpha} V_{i,j}^{n} - (\Delta_h V^{n})_{i,j} + g(x_{i,j},[D_hV^{n}]_{i,j}) \ge  f_h[M^{n+1}]_{i,j},\quad n=0,\dots,N-1,\end{aligned}$$ then $U^n\le V^n$ for any $n=0,\dots,N$.

The existence and uniqueness of $U^{n}$, $n=0, \dots, N-1$ is obtained by induction: at each step, we use Lemma \[le:existence\] with $$\delta = \frac{1}{\rho_\alpha},\qquad V_{i,j} =  \frac{1}{\rho_\alpha} \sum_{k=n+1}^{N} \bar c_{n}^k U_{i,j}^k + f_h[M^{n+1}]_{i,j}.$$ The second part of the statement follows immediately by a comparison principle 