Christian Klingenberg, University of Wuerzburg
601 Pao Yue-Kong Library
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. - Finally we present current work in progress on how to generalize this for the non-linear Euler equations.