Anwendung der "Method of Moving Frames" auf die magnetohydrodynamische Flachwasserströmung - Erhaltungseigenschaften und Robustheit
Mathematik
Zusammenfassung der Projektergebnisse
In this project we considered shallow water equations. One can formulate a general (nonlinear) framework for such equations, however, at this stage, we focused on the linear treatment of the 2D shallow water dynamics. The linearity explicitly implies that we consider a small perturbation over certain equilibrium (the equilibrium here is meant only with respect to hydrodynamics) configurations and that those perturbations of the physical quantities are small compared to the their equilibrium counterparts, that feedback on the equilibrium state could be neglected. The case of shallow water equations is special in a sense that the perturbations of the fluid (plasma) density studied are excluded, leading to the intrinsic incompressible nature of the wave dynamics. In other words, propagation of the density variability, related to the pressure oscillations (acoustic waves), is filtered out from the system. Another property of our set up is that we exclude viscosity and all other transport processes from the equations (ideal (M)HD), meaning absence of dissipation in the system. This is the simplest case and implies the presence of a larger class of intrinsic symmetries in the system, but chosen to demonstrate the concept of moving frames. So, if we can construct the invariant discretization framework, this fulfills the requirements of conservation of a certain number of physical quantities. Schemes which are not invariant under the entire symmetry group of our model equations, but rather only under certain subgroups have been also considered. This is reasonable since not all symmetries of differential equations have the same importance from a physical point of view. In the moving frame method, a novel feature is the consideration of infinite-dimensional Lie symmetry (pseudo)groups. Although the technique of moving frames is readily applicable to such groups, the transfer of this method to the invariantization of numerical schemes is still missing. Starting points for an invariantization has been constituted by rather classical schemes, such as some central difference methods or schemes that preserve some conservation laws numerically. • Mathematical formulation of the MHD shallow water equations and determination of the symmetry group and set of the conserved quantities; • Development of discretization model for the HD shallow-water equations using the method of moving frames; • Development of discretization model for the MHD shallow-water equations using the method of moving frames; • Implementation and testing of the developed numerical scheme in the finite difference framework. • Looking for and establishment of other, additional directions of the collaboration which would be based on the competence of the visiting professor (astrophysics) and host institution (high presision simulations of the comressible flow patterns and turbulence). The results we obtained within the framework of this short initiation of our collaboration were adequate regarding the objectives and consistently addressed the questions we stated. In the case of shallow water equations we were able to proof that the application of the moving frame approach for the discretization gives relatively moderate improvements of the efficiency of the invariant code, we examined, compared to the non-invariant versions of the discretizations.
Projektbezogene Publikationen (Auswahl)
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2020, A new class of discontinuous solar wind solutions, Monthly Notices of the Royal Astronomical Society, 496, 1023
Shergelashvili, Bidzina M.; Melnik, Velentin N.; Dididze, Grigol; Fichtner, Horst; Brenn, Günter; Poedts, Stefaan; Foysi, Holger; Khodachenko, Maxim L.; Zaqarashvili, Teimuraz V.
