Vortrag im Oberseminar Numerik und Optimierung am 21. November 2018
Im Rahmen des Oberseminares "Numerik und Optimierung" hält
Herr Prof. Dr. Christian Klingenberg (Universität Würzburg)
am Mittwoch, 21. November 2018, einen Vortrag mit dem Titel
"Low Mach number asymptotic preserving finite volume methods for the acoustic and the Euler equations in multiple space dimensions"
Beginn: 16:30 Uhr, Raum 25.22.02.81
Alle Interessierten sind herzlich willkommen!
Finite volume methods for compressible Euler equations suffer from an excessive diffusion in the limit of low Mach numbers. This lecture explores approaches to overcome this.
We begin with the the acoustic equations obtained as a linearization of the Euler equations. The limit of compressible to incompressible flows is characterized by a divergence-free velocity. Also in multiple space dimensions advection and acoustics are genuinely different. These concepts are found to have a counterpart in the discrete setting and to be at the origin of the difficulties at resolving the low Mach number limit. We call numerical schemes whose discrete stationary states discretize all the analytic stationary states of the PDE 'stationarity preserving'. It is shown that for the acoustic equations, stationarity preserving schemes are also vorticity preserving and are also asymptotic preserving for the Mach number going to zero. This establishes a new link between these three concepts. We identify all those stencils that are discretizations of the divergence that allow for stabilizing stationarity preserving diffusion. This way we are able to characterize all stable discretization of the acoustic equation that are asymptotic preserving for low Mach numbers.
Stationarity preservation can be generalized to the nonlinear Euler equations. We present numerical schemes for the Euler equations that are stationarity preserving. This leads to a low Mach number asymptotic preserving numerical scheme. Its diffusion is chosen such that it depends on the velocity divergence rather than the derivatives of the different velocity components. This is demonstrated to overcome the low Mach number problem. The scheme shows satisfactory results in numerical simulations and it has been found to be stable under explicit time integration.