## 4 The Linear Wave Equation  
It is well known that the linear wave equation contains second order partial derivatives of the unknown wave function \(u(x,t)\) with respect to time and space, respectively (see, e.g., [20; 21]):  
\[-u_{tt}+u_{xx}=0, \tag{15}\]  
where \(u_{t}\) and \(u_{x}\) denote the derivatives \(\partial u/\partial t\) and \(\partial u/\partial x\), respectively. This equation describes one-dimensional (1D) waves--e.g., the longitudinal string oscillations propagating along the x-axis. The corresponding Lagrangian is [15; 16]  
\[L(u,u_{t},u_{x})=\frac{1}{2}\Big{(}u_{t}^{2}-u_{x}^{2}\Big{)}. \tag{16}\]  
By adopting the triangle discretisation described in Section 3, one gets the discrete Lagrangian  
\[L_{d}\Big{(}u_{i}^{j},u_{i}^{j+1},u_{i+1}^{j+1}\Big{)}=\frac{1}{2}\Delta t \Delta x\left[\frac{1}{2}\left(\frac{u_{i}^{j+1}-u_{i}^{j}}{\phi(\Delta t)} \right)^{2}-\frac{1}{2}\left(\frac{u_{i+1}^{j+1}-u_{i}^{j+1}}{\psi(\Delta x)} \right)^{2}\right], \tag{17}\]  
where \(\Delta t\) and \(\Delta x\) can be thought of as the mesh lengths for time and space, respectively, as stated in Section 3.  
The insertion of the discrete Lagrangian \(L_{d}\) into the discrete Euler-Lagrange field Equation (14) yields the discrete version of (15); i.e.,  
\[-\frac{u_{i}^{j+1}-2u_{i}^{j}+u_{i}^{j-1}}{\left[\phi(\Delta t)\right]^{2}}+ \frac{u_{i+1}^{j}-2u_{i}^{j}+u_{i-1}^{j}}{\left[\psi(\Delta x)\right]^{2}}=0. \tag{18}\]  
The present example was tested on the space domain bounded as \(-10<x<10\), with initial conditions  
\[u_{t}(x,0) = 0,\] \[u(x,0) = sech(x),\]  
and boundary conditions  
\[u(0,t) = u(1,t),\] \[u_{x}(-10,t) = u_{x}(10,t). \tag{19}\]  
The chosen grid discretisation is \(\Delta t=0.1\) and \(\Delta x=0.2\). The resulting solution \(u(x,t)\) is shown in Figure 2 by a 3D diagram. It can be seen that the time evolution of the solution \(u(x=const.,t)\) is a continuous function in which the periodicity is preserved.  
As a second test and for the sake of comparison with the standard multisymplectic methods, we examine the energy behaviour in both \(x\)-directions. Firstly, we considered the temporal energy error (deviation from the initial value) at a given spatial point \(x\) via  
\[e_{j} = \frac{1}{2}\sum_{i=0}^{N-1}\left[\left(\frac{u_{i}^{j+1}-u_{i}^{j }}{\phi(\Delta t)}\right)^{2}+\left(\frac{u_{i+1}^{j+1}-u_{i}^{j+1}}{\psi( \Delta x)}\right)^{2}\right]\Delta x. \tag{20}\]  
Secondly, we calculated the spatial energy error at a given time instant \(t\) via  
\[e_{i} = \frac{1}{2}\sum_{j=0}^{M-1}\left[\left(\frac{u_{i}^{j+1}-u_{i}^{j }}{\phi(\Delta t)}\right)^{2}+\left(\frac{u_{i+1}^{j+1}-u_{i}^{j+1}}{\psi( \Delta x)}\right)^{2}\right]\Delta t. \tag{21}\]  
The obtained results are plotted in Figure 3 and are compared with those of the standard multisymplectic scheme of [11]. The graphs of this figure represent relative errors, i.e., differences from the initial values. While during the numerical simulation, no energy loss or energy blow-up appeared to occur in the system in either case, the proposed scheme improved both energy errors dramatically--the spatial and the temporal ones.  
Figure 2: Calculated waveforms of the linear wave Equation (15).