Gregor Gassner
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MATHEMATICS COLLOQUIUM
Speaker: Gregor Gassner Abstract. When approximating nonlinear PDEs such as the compressible Navier-Stokes equations on discrete nodal grids, aliasing due to interpolation of nonlinear fluxes onto these grids can cause large errors which may even drive instabilities. An interesting approach to decrease the negative effect of this aliasing used primarily in the finite difference community is to reformulate the nonlinearities in different, but equivalent forms. Instead of using the conservative form of the PDE it is for instance possible to use the advective form by applying the chain rule. It is also possible to use an arbitrary mix between those two formulations and introduce skew-symmetric split formulations. By careful choice of such a reformulation it is possible to enhance the stability of finite difference discretisations. Many questions arise, especially if the resulting method is still fully conservative, although it is based on non-conservative forms of the underlying PDEs. In this talk, we show how to incorporate the idea of different split forms of the compressible Navier-Stokes equations into the discontinuous Galerkin ansatz. We show how to recover famous splitting formulations such as the Ducros splitting or the Kennedy and Gruber splitting. We furthermore show that these novel DG schemes are still fully conservative and that with this special formulations the nonlinear stability for unresolved flows such as turbulence is highly enhanced. We will also show ]that the Kennedy and Gruber based splitting is special, as it guarantees consistency with the kinetic energy. Furthermore, this novel approach allows us for the first time to construct a dissipation free DG approximation for the compressible Navier-Stokes equations. The theoretical findings are demonstrated with investigations of the Taylor-Green vortex at different Reynolds numbers. |