**1. Introduction**

In simulations of fluid motion for engineering scenarios, we generally need to deal with obstacles or containment of a non-trivial shape. In mesh-based schemes, we have two options to represent such geometries: we can try to align the mesh with the geometries, such that the walls build a boundary of the mesh or we try to embed the boundary conditions inside the mesh elements. The first option eases the formulation of boundary conditions and their treatment in the scheme [1]. The second option avoids the need to adapt the mesh to the, possibly complex, geometry [2]. Correctly aligning the mesh with arbitrary geometries in the first option can become cumbersome for high-order approximations. Thus, the embedding method is attractive for high-order schemes. Another application area, where

the embedded boundaries provide a benefit, are moving geometries, as the need for new meshes can be avoided during simulations.

High-order discretization schemes can represent smooth solutions with few degrees of freedom. This is an essential property for algorithms on modern computing systems as the memory bandwidth is a strongly limiting factor on new systems, due to the widening memory gap. A numerical scheme that allows for high-order approximations of the solution is the discontinuous Galerkin finite element method. In this method, the solution within elements is represented by a function series (usually a polynomial series). In this work, we are concerned with a high-order discontinuous Galerkin scheme and the embedded geometry representation within it. Besides the possibility to use high-order approximations, the discontinuous Galerkin scheme also offers a relatively loose coupling between elements, resulting in a high computational locality, which in turn is advantageous for modern parallel computing systems. Discontinuous Galerkin methods are, therefore, increasingly popular and relevant.

Peskins [3] was one of the first scientists trying to impose immersed boundaries for his investigations. For his studies, he simulated the flow around heart valves considering the incompressible Navier–Stokes equations while introducing the immersed boundaries, using an elastic model and applying forces to the fluid, thus changing the momentum equations. His work was extended by Saiki and Biringen [4], and they considered feedback forces for the immersed boundaries to represent a rigid body while using an explicit time-stepping, hence resulting in stiff problems and very small time-stepping for the simulation. An important fact, which makes immersed boundary methods more attractive, is the introduction of the effect of the geometry in the governing equations themselves. Embedding the boundaries in the mesh relaxes the requirements on the elements, and using simple elements allows for efficient numerical algorithms that can, for example, exploit inherent symmetric properties of the discretization. The additionally introduced terms can either be considered in the numerical discretization or the continuous equations. Applying forcing terms in the discretization allows for better control of the numerical accuracy and the conservation properties of the used discretization method; on the other hand, the generality and flexibility of these methods disappear when considering different solvers using different discretization methods. In contrast, the volume penalization method imposes additional forcing penalty terms on the continuous equations, while the discretization is done as usual [5]. The Brinkman Penalization Method (BPM) is one of these methods. It was originally developed by Arquis and Caltagirone [6] for numerical simulations of isothermal obstacles in incompressible flows. The idea is to model the obstacle as a porous material, with material properties approaching zero. The major benefit of this method is error estimation, which can be rigorously predicted in terms of the penalization parameters [7]. Furthermore, the boundary conditions can be enforced to a precision, without changing the numerical discretization of the scheme. Kevlahan and Ghidaglia already applied this method for incompressible flows, while considering a non-moving, as well as a moving geometry. They used a pseudo-spectral method [8] in their works.

Liu and Vasilyev employed the volume penalization for the compressible Navier–Stokes equations. In their publication [9], they discussed a 1D and a 2D test case. They used a wavelet method for the discretization and showed error convergence and resulting pressure perturbations for acoustic setups. In other investigations, various numerical discretization methods were used, which showed promising results using the Brinkman penalization method. In [10,11], the pseudo-spectral methods, in [9], wavelet, and in [12], the finite volume/finite element methods were used. However, as far as we know, no work on this kind of penalization in the context of high-order discontinuous Galerkin methods for compressible Navier–Stokes equations has been done so far. Thus, this paper will look into the Brinkman penalization employed within a high-order discontinuous Galerkin solver. Our implementation is available in our open-source solver Ateles [13].
