Methods for higher order numerical simulations of complex inviscid fluids with immersed boundaries

• Main Author: Müller, Björn
• Format: Others
• Language: German; en
• Published: 2014
• Online Access: https://tuprints.ulb.tu-darmstadt.de/3747/1/thesis.pdf; Müller, Björn <http://tuprints.ulb.tu-darmstadt.de/view/person/M=FCller=3ABj=F6rn=3A=3A.html> (2014): Methods for higher order numerical simulations of complex inviscid fluids with immersed boundaries.Darmstadt, Technische Universität, [Ph.D. Thesis]

Within this thesis, we study inviscid compressible flows of fluids modelled by several equations of state. Namely, these are the ideal gas law, the stiffened gas law, Tait's law and the covolume gas law. In their entirety, these equations of state can be used as models for the behaviour of many gases and liquids. After deriving new exact solutions for the corresponding variants of the Euler equations, we use the results as a tool for the verification of a higher-order accurate numerical scheme that has been implemented during the course of this thesis. The scheme is based on a Runge-Kutta Discontinuous Galerkin Method and the presented verification results show that we are able to obtain the expected rates of convergence in both, space and time. In the main part of this thesis, we consider an important building block for the extension of this conventional discretization by means of a treatment for generic immersed boundaries, namely the numerical integration of general functions over domains that are at least partly defined by the zero iso-contour of a level set function defining the domain boundary. Here, we study two new, generally applicable approaches in terms of their robustness and convergence behaviour. The first approach is based on a classical adaptive strategy, while the second approach is based on a hierarchical moment-fitting strategy with variable Ansatz order P. Both methods have been designed such that they are applicable on general element types. Most notably, the results of our numerical experiments suggest that the moment-fitting procedure leads to integration errors that decrease with a rate of O(h^(P+1)), thus allowing for a severe increase of integration accuracy at constant computational effort.