## 3 Discrete Lagrangian Density  
By mimicking the smooth case and following [11; 15; 16], the discrete Lagrangian density can readily be figured out. Towards this purpose, we consider fields over a higher-dimensional oriented manifold \(M\), together with its tangent bundle \(TM\) and cotangent bundle \(T^{*}M\). Then, the Lagrangian density \(L\), is a smooth bundle map over \(X\); i.e.,  
\[L:J^{1}(Y)\rightarrow\Lambda^{n+1}(X), \tag{8}\]  
where \(J^{1}(Y)\) is the first jet bundle over \(Y\) defined via the fibre bundle \(\pi_{XY}:Y\to X\) over the manifold \(X\). In the latter equation, \(\Lambda^{n+1}(X)\) represents the bundle of \((n+1)\)-forms on \(X\) (for more details, see [11] and references therein). Next, we consider the space-time discretisation of \(X\) when \(\dim(X)=2\), which generalises the Veselov discretisation [3; 4] in the multisymplectic field theory. In such a case, the discrete representation of the manifold can be considered as \(X=\mathbb{Z}\times\mathbb{Z}=(i,j)\). On the other hand, the fibre bundle \(Y\) will obey the standard definition of the smooth case, and hence it will come out of the \(X\times F\) for some smooth manifold \(F\)[11; 15; 16]. In this work, we restrict the discretisation to a uniform quadrangular mesh in the base space, with mesh lengths \(\Delta x\) and \(\Delta t\). The nodes in this mesh, which are objects in \(\mathbb{Z}\times\mathbb{Z}\), can be denoted by \((i,j)\) and correspond to points \((x_{i},t_{j}):=(i\Delta x,j\Delta t)\in\mathbb{R}^{2}\). Using the symbol \(u\) for the field values, we denote by \(u_{i}^{j}\) its value at the node \((i,j)\). Then, the triplet \(((i,j),(i+1,j),(i,j+1))\) forms a triangle at the \((i,j)\) which in our notation is denoted by \(\triangle_{ij}\). Finally, the \(X_{\triangle}\) will denote the set of all such triangles; see Figure 1.  
After the above explainations, the discrete jet bundle can be defined as [11; 15; 16]  
\[J^{1}_{\triangle}Y:=\{(u_{i}^{j},u_{i+1}^{j},u_{i}^{j+1})\in\mathbb{R}^{3}:(( i,j),(i+1,j),(i,j+1))\in X_{\triangle}\}, \tag{9}\]  
which is equivalent to \(X_{\triangle}\times\mathbb{R}^{3}\). We remind the reader that an arbitrary value of \(u\) is assumed to be an averaging of the field over the triangle vertices (see Figure 1); i.e.,  
\[u\rightarrow\frac{u_{i}^{j}+u_{i}^{j+1}+u_{i+1}^{j+1}}{3}, \tag{10}\]  
Figure 1: Illustration of spacetime discretisation: triangles at spacetime nodes. and the derivatives of \(u\) are expressed by  
\[\frac{du}{dt}\to\frac{u_{i}^{j+1}-u_{i}^{j}}{\phi(\Delta t)},\qquad\qquad\frac{du} {dx}\to\frac{u_{i+1}^{j+1}-u_{i}^{j+1}}{\psi(\Delta x)}, \tag{11}\]  
(\(\Delta t\) and \(\Delta x\) are the time step and spatial step sizes, respectively) where \(\phi(\Delta t)\) and \(\psi(\Delta x)\) involve the type of the numerical scheme. Specifically, the selection \(\phi(\Delta t)=\Delta t\) and \(\psi(\Delta x)=\Delta x\) leads to the midpoint rule and the generated schemes can be found in [15; 16]. Further, the selection of \(\phi(\Delta t)=2\sin\left(\frac{\Delta t}{2}\right)\) and \(\psi(\Delta x)=2\sin\left(\frac{\Delta x}{2}\right)\) leads to the non-standard finite differences, and the schemes obtained can be found in [17].