@article{Pirner2021,
  author    = {Pirner, Marlies},
  title     = {A review on BGK models for gas mixtures of mono and polyatomic molecules},
  series = {Fluids},
  volume    = {6},
  journal   = {Fluids},
  number    = {11},
  issn      = {2311-5521},
  doi       = {10.3390/fluids6110393},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-250161},
  year      = {2021},
  abstract  = {We consider the Bathnagar-Gross-Krook (BGK) model, an approximation of the Boltzmann equation, describing the time evolution of a single momoatomic rarefied gas and satisfying the same two main properties (conservation properties and entropy inequality). However, in practical applications, one often has to deal with two additional physical issues. First, a gas often does not consist of only one species, but it consists of a mixture of different species. Second, the particles can store energy not only in translational degrees of freedom but also in internal degrees of freedom such as rotations or vibrations (polyatomic molecules). Therefore, here, we will present recent BGK models for gas mixtures for mono- and polyatomic particles and the existing mathematical theory for these models.},
  language  = {en}
}
@phdthesis{Warnecke2022,
  author    = {Warnecke, Sandra},
  title     = {Numerical schemes for multi-species BGK equations based on a variational procedure applied to multi-species BGK equations with velocity-dependent collision frequency and to quantum multi-species BGK equations},
  edition   = {1. Auflage},
  publisher = {W{\"u}rzburg University Press},
  address   = {W{\"u}rzburg},
  isbn      = {978-3-95826-192-1},
  doi       = {10.25972/WUP-978-3-95826-193-8},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-282378},
  school      = {W{\"u}rzburg University Press},
  pages     = {xiii, 203},
  year      = {2022},
  abstract  = {We consider a multi-species gas mixture described by a kinetic model. More precisely, we are interested in models with BGK interaction operators. Several extensions to the standard BGK model are studied. Firstly, we allow the collision frequency to vary not only in time and space but also with the microscopic velocity. In the standard BGK model, the dependence on the microscopic velocity is neglected for reasons of simplicity. We allow for a more physical description by reintroducing this dependence. But even though the structure of the equations remains the same, the so-called target functions in the relaxation term become more sophisticated being defined by a variational procedure. Secondly, we include quantum effects (for constant collision frequencies). This approach influences again the resulting target functions in the relaxation term depending on the respective type of quantum particles. In this thesis, we present a numerical method for simulating such models. We use implicit-explicit time discretizations in order to take care of the stiff relaxation part due to possibly large collision frequencies. The key new ingredient is an implicit solver which minimizes a certain potential function. This procedure mimics the theoretical derivation in the models. We prove that theoretical properties of the model are preserved at the discrete level such as conservation of mass, total momentum and total energy, positivity of distribution functions and a proper entropy behavior. We provide an array of numerical tests illustrating the numerical scheme as well as its usefulness and effectiveness.},
  subject      = {Kinetische Gastheorie},
  language  = {en}
}
@article{HaackHauckKlingenbergetal.2021,
  author    = {Haack, J. and Hauck, C. and Klingenberg, C. and Pirner, M. and Warnecke, S.},
  title     = {A Consistent BGK Model with Velocity-Dependent Collision Frequency for Gas Mixtures},
  series = {Journal of Statistical Physics},
  volume    = {184},
  journal   = {Journal of Statistical Physics},
  number    = {3},
  issn      = {1572-9613},
  doi       = {10.1007/s10955-021-02821-2},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-269146},
  pages     = {31},
  year      = {2021},
  abstract  = {We derive a multi-species BGK model with velocity-dependent collision frequency for a non-reactive, multi-component gas mixture. The model is derived by minimizing a weighted entropy under the constraint that the number of particles of each species, total momentum, and total energy are conserved. We prove that this minimization problem admits a unique solution for very general collision frequencies. Moreover, we prove that the model satisfies an H-Theorem and characterize the form of equilibrium.},
  language  = {en}
}
@phdthesis{Pirner2018,
  author    = {Pirner, Marlies},
  title     = {Kinetic modelling of gas mixtures},
  edition   = {1. Auflage},
  publisher = {W{\"u}rzburg University Press},
  address   = {W{\"u}rzburg},
  isbn      = {978-3-95826-080-1 (Print)},
  doi       = {10.25972/WUP-978-3-95826-081-8},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-161077},
  school      = {W{\"u}rzburg University Press},
  pages     = {xi, 222},
  year      = {2018},
  abstract  = {This book deals with the kinetic modelling of gas mixtures. It extends the existing literature in mathematics for one species of gas to the case of gasmixtures. This is more realistic in applications. Thepresentedmodel for gas mixtures is proven to be consistentmeaning it satisfies theconservation laws, it admitsanentropy and an equilibriumstate. Furthermore, we can guarantee the existence, uniqueness and positivity of solutions. Moreover, the model is used for different applications, for example inplasma physics, for fluids with a small deviation from equilibrium and in the case of polyatomic gases.},
  subject      = {Polyatomare Verbindungen},
  language  = {en}
}