Well-balanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension

Laurent Gosse

doi:10.3934/krm.2012.5.283

Article Contents

2012, Volume 5, Issue 2: 283-323. Doi: 10.3934/krm.2012.5.283

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Well-balanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension

Laurent Gosse^1,

1.
Istituto per le Applicazioni del Calcolo (sezione di Bari), via G. Amendola, 122/D, 70126 BARI

Received: March 31, 2011

Revised: October 31, 2011

Published: May 2012

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Abstract

In the kinetic theory of gases, a class of one-dimensional 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 integro-differential equation whereas the heat transfer is described by a $2 \times 2$ coupled system. This simplification allows to set up the well-balanced method, involving non-conservative 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 steady-state. Boundary-value 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 time-splitting method is displayed for a Couette flow by inspecting the shear stress which must be a constant at steady-state. Other test-cases 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 test-case 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 steady-state free from spurious oscillations is observed.

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Mathematics Subject Classification: Primary: 65M06, 35L65; Secondary: 82B40, 82C40.

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