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Manuscript Title: Nonlinear Boltzmann equation for the homogeneous isotropic case: Minimal deterministic Matlab program
Authors: Pietro Asinari
Program title: HOMISBOLTZ
Catalogue identifier: AEGN_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 181(2010)1776
Programming language: Tested with Matlab version ≥ 6.5. However, in principle, any recent version of Matlab or Octave should work.
Computer: All supporting Matlab or Octave.
Operating system: All supporting Matlab or Octave.
RAM: 300 MBytes
Keywords: Boltzmann equation, homogeneous, isotropic, deterministic method.
PACS: 47.11.-j, 05.20.Dd.
Classification: 23.

Nature of problem:
The problem consists in integrating the homogeneous Boltzmann equation for a generic collisional kernel in case of isotropic symmetry, by a deterministic direct method. Difficulties arise from the multi-dimensionality of the collisional operator and from satisfying the conservation of particle number and energy (momentum is trivial for this test case) as accurately as possible, in order to preserve the late dynamics.

Solution method:
The solution is based on the method proposed by Aristov [1], but with two substantial improvements: (a) the original problem is reformulated in terms of particle kinetic energy (this allows one to ensure exact particle number and energy conservation during microscopic collisions) and (b) a DVM-like correction (where DVM stands for Discrete Velocity Model) is adopted for improving the relaxation rates (this allows one to satisfy exactly the conservation laws at macroscopic level, which is particularly important for describing the late dynamics in the relaxation towards the equilibrium). Both these corrections make possible to derive very accurate reference solutions for this test case.

Restrictions:
The nonlinear Boltzmann equation is extremely challenging from the computational point of view, in particular for deterministic methods, despite the increased computational power of recent hardware. In this work, only the homogeneous isotropic case is considered, for making possible the development of a minimal program (by a simple scripting language) and allowing the user to check the advantages of the proposed improvements beyond Aristov's method [1]. The initial conditions are supposed parameterized according to a fixed analytical expression, but this can be easily modified.

Running time:
From minutes to hours (depending on the adopted discretization of the kinetic energy space). For example, on a 64 bit workstation with Intel CoreTM i7-820Q Quad Core CPU at 1.73 GHz and 8 MBytes of RAM, the provided test run (with the corresponding binary data file storing the pre-computed relaxation rates) requires 154 seconds.

References:
[1] V.V. Aristov, Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows, Kluwer Academic Publishers, 2001.