## 3 Numerical Framework
### Discontinuous Galerkin Discretization
In order to overcome the issue, both SDRS and ERS rely on an artificial compressibility perturbation of (6), which, taking inspiration from the artificial compressibility model by [PERSON] [33], consists of adding a pressure time derivative to the first equation. The set of \(\mathbf{n}\)-split equations for the SDRS reads  
\[ \begin{split}\frac{1}{\rho_{0}a_{0}^{2}}\frac{\partial p}{ \partial t}+\frac{\partial u_{n}}{\partial x_{n}}&=0,\\ \frac{\partial}{\partial t}(\rho u_{n})+\frac{\partial}{\partial x _{n}}\Big{(}\rho u_{n}^{2}+p\Big{)}&=0,\\ \frac{\partial u_{\theta}}{\partial t}+u_{n}\frac{\partial u_{ \theta}}{\partial x_{n}}&=0,\\ \frac{\partial\rho}{\partial t}+u_{n}\frac{\partial\rho}{ \partial x_{n}}&=0,\end{split}  \tag{8}\]  
while the set of \(\mathbf{n}\)-split equations for the ERS reads  
\[ \begin{split}\frac{1}{\rho_{0}a_{0}^{2}}\frac{\partial p}{ \partial t}+\frac{\partial u_{n}}{\partial x_{n}}&=0,\\ \frac{\partial}{\partial t}(\rho u_{n})+\frac{\partial}{\partial x _{n}}\Big{(}\rho u_{n}^{2}+p\Big{)}&=0,\\ \frac{\partial u_{\theta}}{\partial t}+u_{n}\frac{\partial u_{ \theta}}{\partial x_{n}}&=0,\\ \frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{n}} (\rho u_{n})&=0,\end{split}  \tag{9}\]  
where \(\rho_{0}\) is the reference density and \(a_{0}\) is the artificial compressibility parameter. Following [22], we set \(a_{0}\) equal to the reference velocity employed in the definition of the non-dimensional groups, i.e., a unitary value of the dimensionless \(a_{0}\). According to the third equation in (8) and (9), the tangential component of the velocity \(\mathbf{u}_{\theta}\) is purely advected along the normal direction. Since this equation is just the combination of the third and fourth equations of system (6), it does not represent a modification of the original set of \(\mathbf{n}\)-split equations. Moreover, we underline that SDRS enforces the divergence constraint within the continuity equation, thus treating the density as an advected property along the normal direction. The solution of the two Riemann problems (8), (7) and (9), (7), obtained by coupling the artificial compressibility perturbations of (8) with discontinuous initial data, consists of four constant states separated by two external acoustic waves and a central contact discontinuity, see Figure 1.  
Depending on the external states (7), the acoustic waves can be either rarefactions or shocks, thus entailing a change of all flow properties that are not purely advected.The intermediate region embraced by the acoustic waves, denoted as the star region [25], is split into a left (\(\star L\)) and a right (\(\star R\)) portion by the contact discontinuity, across which pressure and normal velocity \(u_{n}\) are constant. We refer to Appendixes A and B of [22] for a complete derivation of the solution for the SDRS and the ERS. Interestingly, while the former requires solving a nonlinear equation system, the latter admits an explicit solution, resulting in better computational efficiency. As a peculiar feature of SDRS we mention that, since the density is purely advected, the density solution at \(\mathcal{S}=0\) is equal either to the left or the right initial values. Accordingly, the density solution is guaranteed to never exceed the density values of the external states (7). As will be pointed out in Section 3.3, this latter feature is of crucial importance when using the Riemann solvers in combination with specific working variables \(\mathbf{w}\) choices.