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Fundamental Principles of the Discontinuous Galerkin Method and Extension to Multiphase Flow Problems

AMS Seminar Series

Speaker: Florian Kummer, Technical University of Darmstadt

Presentation

Seminar Slide Deck (PDF-9.3MB)

Abstract

In recent years, so-called high-order numerical methods have gained increased attention, especially for the application in computational fluid dynamics. This talk focuses on Discontinuous Galerkin (DG) methods, a special subclass of high-order methods that have been applied very successfully to compressible flows but have also achieved attention for incompressible simulations.

The talk will be split into two parts: First, a general introduction into DG methods will be given. The derivation of a DG scheme for a scalar convection equation will be demonstrated and implementation issues will be discussed. This concerns in particular the choice of the polynomial basis with respect to the numerical properties (e.g. convergence order and condition number) as well as the computational performance of the implementation (nodal methods and tensorized basis representation).

In the second part of the talk, an overview over the presenter’s personal contributions to the field is given, mainly on the eXtended Discontinuous Galerkin (XDG) method for the simulation of incompressible two-phase flows. There, the fluid interface is tracked by a level-set method and the shape of the XDG basis functions is dynamically adapted to the position of the interface. Therefore, the spatial approximation space can represent jumps in pressure and velocity gradient “infinitely sharp”. By using this this approach, the high-order convergence observed for the DG method for smooth solutions, can be preserved for the low-regularity, discontinuous solutions, such as those appearing in multi-phase flows.

Biography

Photo of Florian Kummer
Florian Kummer is a research group leader at the Chair of Fluid Dynamics, TU Darmstadt, Germany. He received his Diploma in Mathematics from the University of Innsbruck, Austria (2006) and his Ph.D. (2011) from TU Darmstadt. After a two-year research stay at the Center for Turbulence Research (CTR), Stanford and the Computational and Applied Mathematics (CAAM), Rice University, he returned to Germany. His work is strongly focused on the development of high-order numerical methods for two-phase flows, so-called discontinuous Galerkin methods. Application areas are flows with small Reynolds number and here in particular those with three-phase contact lines. Moreover, he is engaged in viscoelastic flows, chemically reactive flows, and High-Performance-Computing.