Boltzmann Collision Integrals with Mathematica


The website presents the Mathematica source code to evaluate the moments of the Boltzmann collision term as shown in the work:

In this work, we present a methodology to evaluate the moments of the Boltzmann collision term, in a general automated way, using the computer algebra software Mathematica. Based on Grad's distribution function with 26-moments, we compute the non-linear production terms for a simple gas and a granular gas, and the linear production terms for a binary mixture of gases. The results can be obtained for general interaction potential (see below), but in the paper they are given only for hard-sphere interaction potential.

Summary

The pdf-file PaperRGD2012.pdf (AIP Conference Proceedings, Vol. 1501, pp. 67-74) contains the results based on evaluating the production terms with Mathematica.

Program Files

The Mathematica program files can be downloaded through the following zip-file. Please note that the Mathematica codes are not optimized for fastest computation time.

CollisionIntegrals.zip - all program files

The above zip file contains the following files.

  1. EinsteinConvention.m: This file is a Mathematica file, which contains the functions to implement cartesian index notation and Einstein summation convention. It works but still is somewhat experimental. (pdf preview of EinsteinConvention.m).
  2. EinsteinExamples.nb: The file shows some examples on index notation and Einstein summation convention in Mathematica. (pdf preview of EinsteinExamples.nb).
  3. CollisionIntegral.nb: The file shows examples to compute the production terms (for the balance equations of pressure tensor and heat flux) for a simple gas. The results for general interaction potential match with equations (4.42) and (4.43) in the book "An Introduction to the Boltzmann Equation and Transport Processes in Gases" by G. M. Kremer (Springer, 2010). (pdf preview of CollisionIntegral.nb)
  4. SimpleGas.nb: The file contains the Mathematica code for evaluating the non-linear production terms with Grad's 13- and 26-moment distribution function and with different interaction potentials for a simple gas. (pdf preview of SimpleGas.nb)
  5. GranularGas.nb: The file contains the Mathematica code for evaluating the non-linear production terms with Grad's 13- and 26-moment distribution function and with different interaction potentials for a granular gas. (pdf preview of GranularGas.nb)
  6. BinaryMixture.nb: The file contains the Mathematica code for evaluating the linear production terms with Grad's 13- and 26-moment distribution function and with different interaction potentials for a binary mixture of simple gases. (pdf preview of BinaryMixture.nb)

Contact

This website and the code have been provided by Vinay Kumar Gupta and Manuel Torrilhon, both from MATHCCES, RWTH Aachen University. In case of any questions, comments or complaints concerning the code and the method, please feel free to send an email to gupta@mathcces.rwth-aachen.de or mt@mathcces.rwth-aachen.de.



Authors: Vinay Kumar Gupta and Manuel Torrilhon