@phdthesis{Barsukow2018, author = {Barsukow, Wasilij}, title = {Low Mach number finite volume methods for the acoustic and Euler equations}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-159965}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2018}, abstract = {Finite volume methods for compressible Euler equations suffer from an excessive diffusion in the limit of low Mach numbers. This PhD thesis explores new approaches to overcome this. The analysis of a simpler set of equations that also possess a low Mach number limit is found to give valuable insights. These equations are the acoustic equations obtained as a linearization of the Euler equations. For both systems the limit is characterized by a divergencefree velocity. This constraint is nontrivial only in multiple spatial dimensions. As the Jacobians of the acoustic system do not commute, acoustics cannot be reduced to some kind of multi-dimensional advection. Therefore first an exact solution in multiple spatial dimensions is obtained. It is shown that the low Mach number limit can be interpreted as a limit of long times. It is found that the origin of the inability of a scheme to resolve the low Mach number limit is the lack a discrete counterpart to the limit of long times. Numerical schemes whose discrete stationary states discretize all the analytic stationary states of the PDE are called stationarity preserving. It is shown that for the acoustic equations, stationarity preserving schemes are vorticity preserving and are those that are able to resolve the low Mach limit (low Mach compliant). This establishes a new link between these three concepts. Stationarity preservation is studied in detail for both dimensionally split and multi-dimensional schemes for linear acoustics. In particular it is explained why the same multi-dimensional stencils appear in literature in very different contexts: These stencils are unique discretizations of the divergence that allow for stabilizing stationarity preserving diffusion. Stationarity preservation can also be generalized to nonlinear systems such as the Euler equations. Several ways how such numerical schemes can be constructed for the Euler equations are presented. In particular a low Mach compliant numerical scheme is derived that uses a novel construction idea. Its diffusion is chosen such that it depends on the velocity divergence rather than just derivatives of the different velocity components. This is demonstrated to overcome the low Mach number problem. The scheme shows satisfactory results in numerical simulations and has been found to be stable under explicit time integration.}, subject = {Finite-Volumen-Methode}, language = {en} } @phdthesis{Berberich2021, author = {Berberich, Jonas Philipp}, title = {Fluids in Gravitational Fields - Well-Balanced Modifications for Astrophysical Finite-Volume Codes}, doi = {10.25972/OPUS-21967}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-219679}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2021}, abstract = {Stellar structure can -- in good approximation -- be described as a hydrostatic state, which which arises due to a balance between gravitational force and pressure gradient. Hydrostatic states are static solutions of the full compressible Euler system with gravitational source term, which can be used to model the stellar interior. In order to carry out simulations of dynamical processes occurring in stars, it is vital for the numerical method to accurately maintain the hydrostatic state over a long time period. In this thesis we present different methods to modify astrophysical finite volume codes in order to make them \emph{well-balanced}, preventing them from introducing significant discretization errors close to hydrostatic states. Our well-balanced modifications are constructed so that they can meet the requirements for methods applied in the astrophysical context: They can well-balance arbitrary hydrostatic states with any equation of state that is applied to model thermodynamical relations and they are simple to implement in existing astrophysical finite volume codes. One of our well-balanced modifications follows given solutions exactly and can be applied on any grid geometry. The other methods we introduce, which do no require any a priori knowledge, balance local high order approximations of arbitrary hydrostatic states on a Cartesian grid. All of our modifications allow for high order accuracy of the method. The improved accuracy close to hydrostatic states is verified in various numerical experiments.}, subject = {Fluid}, language = {en} } @phdthesis{GallegoValencia2017, author = {Gallego Valencia, Juan Pablo}, title = {On Runge-Kutta discontinuous Galerkin methods for compressible Euler equations and the ideal magneto-hydrodynamical model}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-148874}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2017}, abstract = {An explicit Runge-Kutta discontinuous Galerkin (RKDG) method is used to device numerical schemes for both the compressible Euler equations of gas dynamics and the ideal magneto- hydrodynamical (MHD) model. These systems of conservation laws are known to have discontinuous solutions. Discontinuities are the source of spurious oscillations in the solution profile of the numerical approximation, when a high order accurate numerical method is used. Different techniques are reviewed in order to control spurious oscillations. A shock detection technique is shown to be useful in order to determine the regions where the spurious oscillations appear such that a Limiter can be used to eliminate these numeric artifacts. To guarantee the positivity of specific variables like the density and the pressure, a positivity preserving limiter is used. Furthermore, a numerical flux, proven to preserve the entropy stability of the semi-discrete DG scheme for the MHD system is used. Finally, the numerical schemes are implemented using the deal.II C++ libraries in the dflo code. The solution of common test cases show the capability of the method.}, subject = {Eulersche Differentialgleichung}, language = {en} } @phdthesis{Kanbar2023, author = {Kanbar, Farah}, title = {Asymptotic and Stationary Preserving Schemes for Kinetic and Hyperbolic Partial Differential Equations}, edition = {1. Auflage}, publisher = {W{\"u}rzburg University Press}, address = {W{\"u}rzburg}, isbn = {978-3-95826-210-2}, doi = {10.25972/WUP-978-3-95826-211-9}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-301903}, school = {W{\"u}rzburg University Press}, pages = {xiv, 137}, year = {2023}, abstract = {In this thesis, we are interested in numerically preserving stationary solutions of balance laws. We start by developing finite volume well-balanced schemes for the system of Euler equations and the system of MHD equations with gravitational source term. Since fluid models and kinetic models are related, this leads us to investigate AP schemes for kinetic equations and their ability to preserve stationary solutions. Kinetic models typically have a stiff term, thus AP schemes are needed to capture good solutions of the model. For such kinetic models, equilibrium solutions are reached after large time. Thus we need a new technique to numerically preserve stationary solutions for AP schemes. We find a criterion for SP schemes for kinetic equations which states, that AP schemes under a particular discretization are also SP. In an attempt to mimic our result for kinetic equations in the context of fluid models, for the isentropic Euler equations we developed an AP scheme in the limit of the Mach number going to zero. Our AP scheme is proven to have a SP property under the condition that the pressure is a function of the density and the latter is obtained as a solution of an elliptic equation. The properties of the schemes we developed and its criteria are validated numerically by various test cases from the literature.}, subject = {Angewandte Mathematik}, language = {en} }