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Journal of Statistical Physics

, Volume 170, Issue 1, pp 101–126 | Cite as

On Stable Wall Boundary Conditions for the Hermite Discretization of the Linearised Boltzmann Equation

  • Neeraj Sarna
  • Manuel Torrilhon
Article

Abstract

We define certain criteria, using the characteristic decomposition of the boundary conditions and energy estimates, which a set of stable boundary conditions for a linear initial boundary value problem, involving a symmetric hyperbolic system, must satisfy. We first use these stability criteria to show the instability of the Maxwell boundary conditions proposed by Grad (Commun Pure Appl Math 2(4):331–407, 1949). We then recognise a special block structure of the moment equations which arises due to the recursion relations and the orthogonality of the Hermite polynomials; the block structure will help us in formulating stable boundary conditions for an arbitrary order Hermite discretization of the Boltzmann equation. The formulation of stable boundary conditions relies upon an Onsager matrix which will be constructed such that the newly proposed boundary conditions stay close to the Maxwell boundary conditions at least in the lower order moments.

Keywords

Linearised Boltzmann equaiton Hermite discretization Boundary value problems Stable boundary conditions 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Center for Computational Engineering & Department of Mathematics, Mathematics (CCES)RWTH Aachen UniversityAachenGermany

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