Wasilij Barsukow

Wasilij Barsukow, Philipp V. F. Edelmann, Christian Klingenberg et al.

Based on the Roe solver a new technique that allows to correctly represent low Mach number flows with a discretization of the compressible Euler equations was proposed in Miczek et al.: New numerical solver for flows at various mach numbers. A&A 576, A50 (2015). We analyze properties of this scheme and demonstrate that its limit yields a discretization of the continuous limit system. Furthermore we perform a linear stability analysis for the case of explicit time integration and study the performance of the scheme under implicit time integration via the evolution of its condition number. A numerical implementation demonstrates the capabilities of the scheme on the example of the Gresho vortex which can be accurately followed down to Mach numbers of ~1e-10 .

Wasilij Barsukow

There is a qualitative difference between one-dimensional and multi-dimensional solutions to the Euler equations: new features that arise are vorticity and a nontrivial incompressible (low Mach number) limit. They present challenges to finite volume methods. It seems that an important step in this direction is to first study the new features for the multi-dimensional acoustic equations. There exists an analogue of the low Mach number limit for this system and its vorticity is stationary. It is shown that a scheme that possesses a stationary discrete vorticity (vorticity preserving) also has stationary states that are discretizations of all the analytic stationary states. This property is termed stationarity preserving. Both these features are not generically fulfilled by finite volume schemes; in this paper a condition is derived that determines whether a scheme is stationarity preserving (or, equivalently, vorticity preserving) on a Cartesian grid. Additionally, this paper also uncovers a previously unknown connection to schemes that comply with the low Mach number limit. Truly multi-dimensional schemes are found to arise naturally and it is shown that a multi-dimensional discrete divergence previously discussed in the literature is the only possible stationarity preserving one (in a certain class).

Wasilij Barsukow, Christian Klingenberg, Simon Krotsch

The Active Flux (AF) method employs a globally continuous approximation, like continuous Finite Element methods. This is achieved through the placement of point values at cell interfaces which are shared between adjacent cells. With, on average, K+1 degrees of freedom per cell, Active Flux achieves a polynomial approximation of degree K+1, while the Discontinuous Galerkin (DG) method uses only polynomials of degree K, i.e. one degree less with the same number of degrees of freedom. Despite all the differences, in this paper we show, however, that for linear problems in one and several dimensions as well as -- in some sense -- for nonlinear ones, semi-discrete AF and DG are the same method. We identify a mapping between their respective degrees of freedom, upon which the updates of these degrees of freedom turn out to agree. On the one hand, AF therefore seems more economical then DG for a given value of the error, and we confirm this in numerical experiments. On the other hand, this is a way to understand superconvergence of DG in a natural way, and we show how Radau polynomials and their zeros appear in the mapping between DG and AF: In the Radau points, AF "shines through" as the background high-order scheme behind DG.