Author(s) 
Laurent Gosse ()^{1} 
Laboratory 

Subject 

Title 
Wellbalanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension 
Abstract 
In the kinetic theory of gases, a class of onedimensional problems can be distinguished for which transverse momentum and heat transfer effects decouple. This feature is revealed by projecting the linearized Boltzmann model onto properly chosen directions (which were originally discovered by Cercignani in the sixties) in a Hilbert space. The shear flow effects follow a scalar integrodifferential equation whereas the heat transfer is described by a $2 \times 2$ coupled system. This simplification allows to set up the wellbalanced method, involving nonconservative products regularized by solutions of the stationary equations, in order to produce numerical schemes which do stabilize in large times and deliver accurate approximations at numerical steadystate. Boundaryvalue problems for the stationary equations are solved by the technique of ''elementary solutions" at the continuous level and by means of the ''analytical discrete ordinates" method at the numerical one. Practically, a comparison with a standard timesplitting method is displayed for a Couette flow by inspecting the shear stress which must be a constant at steadystate. Other testcases are treated, like heat transfer between two unequally heated walls and also the propagation of a sound disturbance in a gas at rest. Other numerical experiments deal with the behavior of these kinetic models when the Knudsen number becomes small. In particular, a testcase involving a computational domain containing both rarefied and fluid regions characterized by mean free paths of different magnitudes is presented: stabilization onto a physically correct steadystate free from spurious oscillations is observed. 
Fulltext language 
English 

DOI 
10.3934/krm.2012.5.283 
Journal 
Kinetic and related models 
Audience 
international 
Publication date 
201206 
Volume 
5 
Issue 
2 
Page, identifiant, ... 
283  323 

Keyword(s) 
Slow rarefied flow – Wellbalanced scheme – Elementary solutions – Analytical discreteordinates method – Integral equation of the third kind 
Classification 
Primary 65M06, 35L65; Secondary 82B40,82C40 
