@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}
}