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A Parallel DG Scheme for Computational Fluid Dynamics using Advanced Time Integration Methods

Sascha Truebelhorn, Veronika Diba, and Sigrun Ortleb, Department of Mathematical and Natural Sciences, University of Kassel, Germany,  and Matthias K. Gobbert, Department of Mathematics and Statistics, UMBC

Discontinuous space discretizations, especially the Discontinuous Galerkin(DG) schemes, are a modern and popular class of numerical methods especially for computationally intensive fluid dynamics calculations. Their popularity is due to the fact that DG methods allow for high order approximations in combination with high flexibility – e.g. in choosing different polynomial degrees on neighbouring elements.  In the context of practically relevant problems, semi-discrete DG equations are often extremely stiff. In the case of complex geometries, e.g. for fluid flow around obstacles, the DG mesh is locally refined with elements very different in size. In addition, for high Reynods numbers, applications require a considerable grid refinement in boundary layer zones. In this context, the time integration methods applied so far are yet far from being efficient. Especially with regard to the skillful coupling of explicit and implicit methods, considerably more research is needed. In this project, the application of IMEX and exponential integrators to DG space discretizations of challenging CFD test cases will be studied.
Furthermore, the  challenge of the future in order to enable reliable simulations of complex real life problems is the design of methods for parallel applications. DG methods are perfectly suitable for an implementation within parallel hardware environments. In preparatory work, a robust, high order DG scheme with low numerical dissipation based on novel efficient filtering strategies has been developed. Based on this groundwork, high standard parallel implementation will allow to use this method for example in the context of fluid-structure-interaction. More precisely, the DG method will be used to simulate the air flow around a heated structure, the practical application being thermo-mechanical production processes to obtain graded workpieces.