Research Topic

A high-order discontinuous Galerkin method for unsteady compressible flows with immersed boundaries

In a variety of industrial applications the understanding of compressible flows is of major interest and ranges from the accurate prediction of high speed flows to the deeper understanding of the collapse of a cavitation bubble. In many cases, state of the art tools applied in industry are not able to correctly capture the complex physics due to their low accuracy. Thus, high-order methods gain considerably popularity because of their improved accuracy. One of these methods is the discontinuous Galerkin method. However, there are still numerous unsolved problems in context of high-order methods, for example difficult mesh generation with curved elements, higher memory consumption and inefficient time integration.


Recent work

Scope of this work is the extension of the existing compressible DG solver in the framework BoSSS at the department of fluid dynamics (FDY). In this study, we develop a high-order discontinuous Galerkin method which addresses two of these problems. First, we simplify the mesh generation with the help of cut cells. While common methods demand a boundary conforming mesh, we cut the geometry out of a much simpler background mesh. This approach is also known as immersed boundary method. A quadrature technique based on an hierarchical moment-fitting strategy is proposed for an accurate integration on arbitrarily shaped cut cells. Further, the problems arising from very small cells or strong curved cut cells are solved by introducing a non-intrusive cell agglomerating strategy, which exploits the locality of the discontinuous Galerkin operator. Second, the severe time step restrictions due to anisotropic meshes are remedied by introducing a local time stepping algorithm.


Recent Results

Convergence study for a flow over a Gaussian bump
Flow around a sphere

Various numerical experiments confirm the high accuracy of the proposed scheme. The optimal rate of convergence is obtained in space and time for sufficiently smooth problems, e.g. the flow over a Gaussian Bump (see right). The geometrical flexibility of the immersed boundary method is pointed out by comparing different setups of flows around an airfoil. Further, we present the favorable accuracy of higher order ansatz functions compared to the low order ones in terms of degrees of freedom, even for non-smooth problems. The simulation of a flow around a cylinder at moderate Reynolds number with different Mach numbers confirms excellent agreement with experiments and numerical benchmark results. Even though the computational costs for cut cells are higher than uncut cells, the immersed boundary method is competitive to a boundary fitted approach with curved elements. Finally, the assessment of the local time stepping algorithm reveals a relaxed time step restriction and its runtime is beneficial compared to global time stepping schemes. The presented results demonstrate that the novel immersed boundary method based on a discontinuous Galerkin scheme is well suited to analyze compressible flows of complex geometries, while the mesh generation remains simple. Recently, the complete scheme is extended to 3D problems, e.g., the flow around a sphere (see Figure).


Key Research Area

Compressible flows; Discontinuous Galerkin method; extended/unfitted discretization methods


Stephan Krämer-Eis


Dolivostraße 15

D-64293 Darmstadt



+49 6151 16 - 24383


+49 6151 16 - 24404




kraemer-eis (at) gsc.tu...

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