A continuous interior penalty method for the linear regularized 13-moment equations describing rarefied gas flows

Westerkamp, Armin; Torrilhon, Manuel (Thesis advisor); Van Brummelen, Harald (Thesis advisor)

Aachen (2017, 2018)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2017


The accurate and computationally efficient prediction of rarefied gas flows remains a challenging research topic up to today. Particle methods due to their simplicity in setup and inherent universal applicability state a powerful tool in that regard. Yet, they are demanding with respect to the computational resources and the simulation time grows with the number of particles involved. This has left researchers with a lack of tools in the so-called transition regime, in which classical continuum models, in particular the Navier-Stokes-Fourier equations, fail and particle methods are almost forbiddingly costly. The broader context of this thesis is to highlight how to fill the gap by extending continuum models into the transition regime. The regularized 13-moment (R13) equations, based on the findings of kinetic theory and moment methods, state the underlying model that shall enable this endeavor. This thesis is devoted to the development of the necessary numerical scheme that allows to apply R13 on real-life scenarios. The focus is put on a linearized subsystem in two space dimensions which can be regarded as an important stepping stone towards the full nonlinear system. In the first part of the monograph the theoretical background of the R13 equations is outlined by introducing kinetic theory, moment methods and non-equilibrium thermodynamics. R13 does not state the sole solution of finding suitable continuum models and an assessment of the model in the broader context of moment closures is undertaken. The particular advantages for extending modern CFD by R13 are outlined. Furthermore the topic of solid wall boundary conditions is discussed. The second part then presents the main findings of the underlying research constituting this thesis by outlining the numerical challenges that arise. Special emphasis is put on an instability, that is closely tied to the particular choice of boundary conditions. The instability is analyzed in detail and the gathered knowledge is used to derive a suitable scheme, based on the continuous interior penalty method, for the linearized subsystem of the R13 equations. Then the advantage of applying high order methods is outlined and additionally augmented by a hybridization technique. The solver is then assessed by applying it to a Knudsen pump setup, that not only highlights the solver’s capabilities, but also illustrates the advantage of R13 in comparison to standard hydrodynamic equations.


  • Department of Mathematics [110000]
  • Chair of Applied and Computational Mathematics [115010]