**4. Conclusions**

We showed that, with the help of an implicit mixed explicit time integration approach, it was feasible to implement wall boundaries accurately in a high-order discontinuous Galerkin scheme. The additional source terms introduced for the penalization can be efficiently computed in the implicit part of the mixed time integration without the need for iterative solvers. This implicit treatment enabled us to utilize arbitrary small values for the permeabilities and freed us from the need for the porosity introduced by Liu and Vasiliyev for compressible flows. The viability of the approach was shown in one-dimensional examples, where we saw that the solid wall can be well approximated with small permeabilities in the high-order discontinuous Galerkin scheme. Even for the reflection of a shock wave, for which a high-order discretization is problematic due to the oscillations incurred by the discontinuity, the penalization provided small errors and convergence with higher polynomial degrees. The real strength of the penalization method, however, came through in multiple dimensions, where curved boundaries could easily be represented by the penalization consistent with the scheme. As an example of such a setting, we looked at the acoustic wave scattering at a cylinder.

With the presented method, it, therefore, was possible to exploit the benefit of reduced memory consumption by the high-order discretization even for complex geometries without the need for advanced mesh generation.

**Author Contributions:** Conceptualization by N.A. and H.K.; N.A. wrote the original draft preparation and N.E.P. contributed the shock reflection setup; all authors were involved in the review and editing process; N.A., N.E.P. and H.K. worked on the presented methodology; H.K. and N.E.P. worked on the employed software; investigation and validation was carried out by N.A. and N.E.P., they also did the visualization to produce the graphs and images. supervision, S.R.; funding acquisition, S.R.

**Funding:** Neda Ebrahimi Pour was financially supported by the priority program 1648–Software for Exascale Computing 214 (www.sppexa.de) of the Deutsche Forschungsgemeinschaft.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
