## 4 Numerical Results
### Density Waves  
In this section, the influence of the space and time integration accuracy on the mass conservation property of the proposed approach is analyzed for each change in density variable proposed in Section 3.3 on two very simple inviscid (\(\mu=0\)) test cases:  
* The linear advection of a sine density wave \[\rho=\frac{r_{\rho}+1}{2r_{\rho}}+\frac{r_{\rho}-1}{2r_{\rho}}\sin\left(\frac{ \pi}{5}\dot{x}\right);\]
* The linear advection of a square-density wave \[\rho=\left\{ \begin{array}{ll}1&\left|\frac{\ddot{x}+6}{10}\right|-\left| \frac{\ddot{x}+4}{10}\right|=1\\ {r_{\rho}}^{-1}&\text{otherwise}\end{array} \right. ;\]  
where \(\ddot{x}=x-ut\) is the space-time coordinate, given the dimensionless uniform advection velocity \(u=1\) and \(x\in\Omega=(0,10)\), \(t\in(0,10)\). Non-dimensional pressure is set unitary \(p=1\) over the entire domain \(\Omega\) at \(t=0\) and periodic boundary conditions are imposed at \(x=0\) and \(x=10\). Despite being one-dimensional, both test cases are computed with a two-dimensional solver. Accordingly, the initial condition for the velocity field is \(\mathbf{u}=ut\) with symmetric boundary conditions imposed in span direction \(y\in(0,1)\). The relative mass conservation error is here defined as  
\[err_{\mathcal{M}}=\left|\frac{\mathcal{M}(t_{f})-\mathcal{M}(t_{0})}{\mathcal{ M}(t_{0})}\right|,\]  
where, trivially, the mass is \(\mathcal{M}(t)=\int_{\Omega}\rho(t)\,\mathrm{d}x\) and \(t_{0}=0\). In Figure 3, we represent the relative mass conservation error achieved after one period (\(t_{f}=10\)) of the sine density wave test case with the three different choices of the change of density variable: linear (11), exponential (12) and hyperbolic tangent-based (13), respectively denoted as \(f_{1}(\widetilde{\rho})\), \(f_{2}(\widetilde{\rho})\) and \(f_{3}(\widetilde{\rho})\). The density ratio reads \(r_{\rho}=2\)  
Figure 2: Guermond–Quartapelle manufactured solution—temporal convergence analysis for each variable using the implicit ESDIRK5(8) scheme and the time step adaptation. Space discretization is applied on a uniform grid of \(10\times 1\) quadrangular elements using a DG polynomial approximation of degree \(k=1\) (left) and \(k=6\) (right). Time integration is performed with the ESDIRK5(8) scheme and the time adaptation algorithm. Since the solution is smooth, the oscillation control strategy is not active. The results suggest that the behavior of the mass conservation error is not influenced by the polynomial degree. In particular, for both \(k=1\) and \(k=6\), the linear- and the hyperbolic tangent-based transformations ensure machine precision; in the case of the exponential transformation, the rate of convergence is driven by the temporal scheme accuracy, i.e., 5-th order. In Figure 4, we report the relative mass conservation error obtained after one period (\(t_{f}=10\)) of the square density wave test case by imposing \(r_{\rho}=2\), on the left, and \(r_{\rho}=800\) (water-air density ratio), on the right. Here, space discretization is performed with a DG polynomial approximation of degree six (\(k=6\)) on a uniform \(10\times 1\) quadrangular elements grid. Time integration is carried out based on the ESDIRK5(8) scheme with time step adaption. The artificial diffusion terms for spurious oscillation dampening are active within the domain region where the density is discontinuous. With a small density ratio (\(r_{\rho}=2\) on left), both exponential and hyperbolic tangent changes of density variables show convergent behavior of the mass error for large time steps, again driven by the temporal scheme accuracy, i.e., 5-th order. However, the mass error reaches a plateau when considering smaller step sizes. The linear transformation still ensures mass conservation up to machine precision. For a high-density ratio (\(r_{\rho}=800\) on right), instead, the mass error does not converge with either the \(f_{2}(\widetilde{\rho})\) or \(f_{3}(\widetilde{\rho})\) transformations but remains roughly constant for different \(\overline{\Delta t}\).