@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}
}