@phdthesis{Birke2024,
  author    = {Birke, Claudius B.},
  title     = {Low Mach and Well-Balanced Numerical Methods for Compressible Euler and Ideal MHD Equations with Gravity},
  doi       = {10.25972/OPUS-36330},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-363303},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2024},
  abstract  = {Physical regimes characterized by low Mach numbers and steep stratifications pose severe challenges to standard finite volume methods. We present three new methods specifically designed to navigate these challenges by being both low Mach compliant and well-balanced. These properties are crucial for numerical methods to efficiently and accurately compute solutions in the regimes considered. First, we concentrate on the construction of an approximate Riemann solver within Godunov-type finite volume methods. A new relaxation system gives rise to a two-speed relaxation solver for the Euler equations with gravity. Derived from fundamental mathematical principles, this solver reduces the artificial dissipation in the subsonic regime and preserves hydrostatic equilibria. The solver is particularly stable as it satisfies a discrete entropy inequality, preserves positivity of density and internal energy, and suppresses checkerboard modes. The second scheme is designed to solve the equations of ideal MHD and combines different approaches. In order to deal with low Mach numbers, it makes use of a low-dissipation version of the HLLD solver and a partially implicit time discretization to relax the CFL time step constraint. A Deviation Well-Balancing method is employed to preserve a priori known magnetohydrostatic equilibria and thereby reduces the magnitude of spatial discretization errors in strongly stratified setups. The third scheme relies on an IMEX approach based on a splitting of the MHD equations. The slow scale part of the system is discretized by a time-explicit Godunov-type method, whereas the fast scale part is discretized implicitly by central finite differences. Numerical dissipation terms and CFL time step restriction of the method depend solely on the slow waves of the explicit part, making the method particularly suited for subsonic regimes. Deviation Well-Balancing ensures the preservation of a priori known magnetohydrostatic equilibria. The three schemes are applied to various numerical experiments for the compressible Euler and ideal MHD equations, demonstrating their ability to accurately simulate flows in regimes with low Mach numbers and strong stratification even on coarse grids.},
  subject      = {Magnetohydrodynamik},
  language  = {en}
}
@phdthesis{Maier2008,
  author    = {Maier, Andreas},
  title     = {Adaptively Refined Large-Eddy Simulations of Galaxy Clusters},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-32274},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2008},
  abstract  = {It is aim of this work to develop, implement, and apply a new numerical scheme for modeling turbulent, multiphase astrophysical flows such as galaxy cluster cores and star forming regions. The method combines the capabilities of adaptive mesh refinement (AMR) and large-eddy simulations (LES) to capture localized features and to represent unresolved turbulence, respectively; it will be referred to as Fluid mEchanics with Adaptively Refined Large-Eddy SimulationS or FEARLESS.},
  subject      = {Turbulenz},
  language  = {en}
}
@phdthesis{Schnuecke2016,
  author    = {Schn{\"u}cke, Gero},
  title     = {Arbitrary Lagrangian-Eulerian Discontinous Galerkin methods for nonlinear time-dependent first order partial differential equations},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-139579},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2016},
  abstract  = {The present thesis considers the development and analysis of arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) methods with time-dependent approximation spaces for conservation laws and the Hamilton-Jacobi equations. Fundamentals about conservation laws, Hamilton-Jacobi equations and discontinuous Galerkin methods are presented. In particular, issues in the development of discontinuous Galerkin (DG) methods for the Hamilton-Jacobi equations are discussed. The development of the ALE-DG methods based on the assumption that the distribution of the grid points is explicitly given for an upcoming time level. This assumption allows to construct a time-dependent local affine linear mapping to a reference cell and a time-dependent finite element test function space. In addition, a version of Reynolds' transport theorem can be proven. For the fully-discrete ALE-DG method for nonlinear scalar conservation laws the geometric conservation law and a local maximum principle are proven. Furthermore, conditions for slope limiters are stated. These conditions ensure the total variation stability of the method. In addition, entropy stability is discussed. For the corresponding semi-discrete ALE-DG method, error estimates are proven. If a piecewise \$\mathcal{P}^{k}\$ polynomial approximation space is used on the reference cell, the sub-optimal \$\left(k+\frac{1}{2}\right)\$ convergence for monotone fuxes and the optimal \$(k+1)\$ convergence for an upwind flux are proven in the \$\mathrm{L}^{2}\$-norm. The capability of the method is shown by numerical examples for nonlinear conservation laws. Likewise, for the semi-discrete ALE-DG method for nonlinear Hamilton-Jacobi equations, error estimates are proven. In the one dimensional case the optimal \$\left(k+1\right)\$ convergence and in the two dimensional case the sub-optimal \$\left(k+\frac{1}{2}\right)\$ convergence are proven in the \$\mathrm{L}^{2}\$-norm, if a piecewise \$\mathcal{P}^{k}\$ polynomial approximation space is used on the reference cell. For the fullydiscrete method, the geometric conservation is proven and for the piecewise constant forward Euler step the convergence of the method to the unique physical relevant solution is discussed.},
  subject      = {Galerkin-Methode},
  language  = {en}
}