In this paper, we construct a hierarchy of hybrid numerical methods for multi-scale kinetic equations based on moment realizability matrices, a concept introduced by Levermore, Morokoff, and Nadiga in [Phys. Fluids, 10 (1998), pp. 3214--3226]. Following such a criterion, one can consider a hybrid scheme where the hydrodynamic part is given either by the compressible Euler or Navier--Stokes equations, or even with more general models, such as the Burnett or super-Burnett systems.
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Filbet et al. (2015) studied this question.
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