Progress in Both All Mach Number Flow and Well-Balanced Methods for the Compressible Euler Equations
Speaker: Christian Klingenberg, University of Wuerzburg
Presentation
Abstract
Many astrophysical systems feature flows are modeled by the multi-dimensional Euler equations. We discuss two aspects. First, for the homogeneous Euler equations, we look at flow in the low Mach number regime. Here for conventional finite volume discretizations one has excessive dissipation in this regime. We identify inconsistent scaling with low Mach number of numerical flux function as the origin of this problem. We propose a new flux preconditioner that ensures the correct scaling. We demonstrate that our new method is capable of representing flows down to Mach numbers of 10e-10. This is joint work with Wasilij Barsukow, Philipp Edelmann and Fritz Röpke.
Second, for the Euler equations with gravity, we seek well-balanced methods. We describe a numerical discretization of the compressible Euler equations with a gravitational potential. A pertinent feature of the solutions to these inhomogeneous equations is the special case of stationary solutions with zero velocity, described by a nonlinear PDE, whose solutions are called hydrostatic equilibria. We present a well-balanced method, for which we can ensure robustness, accuracy and stability, since it satisfies discrete entropy inequalities. This is joint work with Wasilij Barsukow, Christophe Berthon, Vivien Desveaux, Philipp Edelmann, Fritz Röpke and Markus Zenk.
Biography
Professor Klingenberg is a mathematician. He obtained his Ph.D. in New York at the Courant Institute, where he continued on as a post-doc. He then moved to Heidelberg, Germany and later became a professor in Wuerzburg, Germany. He will be the co-organizer of the 16th International Conference on Hyperbolic Problems in Aachen, Germany in August of 2016.
In his research he models physical phenomena through both mathematical analysis of PDEs and computational numerical methods. The mathematical models are mainly conservation laws. His core interests are the compressible hydrodynamic and magnetohydrodynamics equations. On the one hand he considers qualitative behavior of their solutions, existence, asymptotic behavior, also for model problems and their derivation from models in statistical physics. On the other hand he works on numerical methods for large scale numerical computations in the context of astrophysics. For more details see his web page.