**3. Results and Discussion**

To investigate the penalization scheme in our discontinuous Galerkin implementation, we first analyzed the fundamental behavior in two one-dimensional setups and then considered the scattering at a cylinder in a two-dimensional setup.

As explained in Section 2.2, the modeling error by the penalization for the compressible Navier–Stokes equations as found by Liu and Vasilyev [9] was expected to scale with the porosity *φ* by an exponent between 3/4 and one and with the viscous permeability *η* by an exponent between 1/4 and 1/2. To achieve low errors, you may, therefore, be inclined to minimize *φ*. However, with the implicit mixed explicit time integration scheme presented in Section 2.4, we can eliminate the stiffness issues due to small permeabilities with little additional costs, while the stability limitation by the porosity persists. Because of this, we deem it more feasible to utilize a small viscous permeability instead of a small porosity. At the same time, the relation between viscous permeability *η* and thermal permeability *η<sup>T</sup>* gets small without overly large *ηT*. Therefore, we used a slightly different scaling than

proposed by Liu and Vasilyev [9]. We introduce the scaling parameter *β* and define the permeabilities accordingly in relation to the porosity as follows.

$$
\eta = \beta^2 \cdot \phi^2 \tag{33}
$$

$$
\eta\_T = 0.4\beta \cdot \phi \tag{34}
$$

Note, that we then expect the modeling error to be of size *O*(*β*1/4*φ*3/4).
