\tilde{M}^{n+1}\bigg)_2 \\
& +   \sum_{n=0}^{N-1}  \sum_{i,j} M_{i,j}^{n+1}R^n_{i,j}(U,{\tilde U})+ \sum_{n=0}^{N-1}  \sum_{i,j} \tilde{M}_{i,j}^{n+1} R^n_{i,j}({\tilde U},U)\\
&=\sum_{n=0}^{N-1}(a^n,M^{n+1}-{\tilde M}^{n+1})_2+\sum_{n=0}^{N-1}(b^{n+1},U^n-{\tilde U}^n)_2
\end{aligned}$$ where $$R^n_{i,j}(V,W)=g(x_{i,j},[D_hW^{n}]_{i,j}) - g(x_{i,j},[D_hV^{n}]_{i,j})   -[D_h(W^{n}-V^{n})]_{i,j} \cdot  g_q  (x_{i,j},[D_hV^{n}])$$ for $V=(V^n)_{0\le n\le N}$, $W=(W^n)_{0\le n\le N}$.

The identity can be obtained by arguing as in the proof of Theorem \[thm:uniqueness\], see in particular , and taking into account the additional terms $a^n_{i,j}$, $b^n_{i,j}$.

For the convergence analysis, we consider the reference case $$\label{eq:conv_ham}
H(x,p)=|p|^\beta+c(x)$$ where $\beta\ge 2$ and $c:{\mathbb{T}}^2\to{\mathbb{R}}$ is a continuous function, but the analysis can be extended in a more general setting. For the approximation of the Hamiltonian, we consider the numerical Hamiltonian $$\label{eq:conv_num_ham}
g(x,q)=((q_1^-)^2+(q_2^+)^2+(q_3^-)^2+(q_4^+)^2)^{\beta/2}+c(x)$$ where $q^\pm$ denote the positive and the negative part of $q\in{\mathbb{R}}$.

Assume that the continuous problem admits a unique classical solution $(u,m)\in C^{2}({\mathbb{T}}^2\times [0,T])\times C^{2}({\mathbb{T}}^2\times [0,T])$ and that $m_0\ge \delta>0$. For ${\sigma}=(h,{{\Delta t}})$, denote by $(u_{\sigma},m_{\sigma})$ the linear function obtained by interpolating the solution $(U,M)$ of on the nodes of the space-time grids. Then, as $|{\sigma}|\to 0$, $(u_{\sigma},m_{\sigma})$ converges to $(u,m)$ in $L^\beta(0,T: W^{1,\beta}({\mathbb{T}}^2))\times  (C(0,T;L^2({\mathbb{T}}^2)) \cap L^2(0,T;H^1({\mathbb{T}}^2))$.

The idea of the proof is to show that the solution of the continuous problem satisfies the perturbed discrete problem with $(a^n,b^n)$ vanishing for the discretion steps ${{\Delta t}},h$ going 