Nonlinear Boltzmann equation for the homogeneous isotropic case: Minimal deterministic Matlab program
The homogeneous isotropic Boltzmann equation (HIBE) is a fundamental dynamic model for many applications in thermodynamics, econophysics and sociodynamics. Despite recent hardware improvements, the solution of the Boltzmann equation remains extremely challenging from the computational point of view,...
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| Vydané v: | Computer physics communications Ročník 181; číslo 10; s. 1776 - 1788 |
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| Jazyk: | English |
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Elsevier B.V
01.10.2010
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| ISSN: | 0010-4655, 1879-2944 |
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| Abstract | The homogeneous isotropic Boltzmann equation (HIBE) is a fundamental dynamic model for many applications in thermodynamics, econophysics and sociodynamics. Despite recent hardware improvements, the solution of the Boltzmann equation remains extremely challenging from the computational point of view, in particular by deterministic methods (free of stochastic noise). This work aims to improve a deterministic direct method recently proposed [V.V. Aristov, Kluwer Academic Publishers, 2001] for solving the HIBE with a generic collisional kernel and, in particular, for taking care of the late dynamics of the relaxation towards the equilibrium. Essentially (a) the original problem is reformulated in terms of particle kinetic energy (exact particle number and energy conservation during microscopic collisions) and (b) the computation of the relaxation rates is improved by the DVM-like correction, where DVM stands for Discrete Velocity Model (ensuring that the macroscopic conservation laws are exactly satisfied). Both these corrections make possible to derive very accurate reference solutions for this test case. Moreover this work aims to distribute an open-source program (called
HOMISBOLTZ), which can be redistributed and/or modified for dealing with different applications, under the terms of the GNU General Public License. The program has been purposely designed in order to be minimal, not only with regards to the reduced number of lines (less than 1000), but also with regards to the coding style (as simple as possible).
Program title: HOMISBOLTZ
Catalogue identifier: AEGN_v1_0
Program summary URL:
http://cpc.cs.qub.ac.uk/summaries/AEGN_v1_0.html
Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland
Licensing provisions: GNU General Public License
No. of lines in distributed program, including test data, etc.: 23 340
No. of bytes in distributed program, including test data, etc.: 7 635 236
Distribution format: tar.gz
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
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 (2001) [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 (2001) 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. |
|---|---|
| AbstractList | The homogeneous isotropic Boltzmann equation (HIBE) is a fundamental dynamic model for many applications in thermodynamics, econophysics and sociodynamics. Despite recent hardware improvements, the solution of the Boltzmann equation remains extremely challenging from the computational point of view, in particular by deterministic methods (free of stochastic noise). This work aims to improve a deterministic direct method recently proposed [V.V. Aristov, Kluwer Academic Publishers, 2001] for solving the HIBE with a generic collisional kernel and, in particular, for taking care of the late dynamics of the relaxation towards the equilibrium. Essentially (a) the original problem is reformulated in terms of particle kinetic energy (exact particle number and energy conservation during microscopic collisions) and (b) the computation of the relaxation rates is improved by the DVM-like correction, where DVM stands for Discrete Velocity Model (ensuring that the macroscopic conservation laws are exactly satisfied). Both these corrections make possible to derive very accurate reference solutions for this test case. Moreover this work aims to distribute an open-source program (called HOMISBOLTZ), which can be redistributed and/or modified for dealing with different applications, under the terms of the GNU General Public License. The program has been purposely designed in order to be minimal, not only with regards to the reduced number of lines (less than 1000), but also with regards to the coding style (as simple as possible). Program title: HOMISBOLTZ Catalogue identifier: AEGN_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEGN_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License No. of lines in distributed program, including test data, etc.: 23a[control]340 No. of bytes in distributed program, including test data, etc.: 7a[control]635a[control]236 Distribution format: tar.gz Programming language: Tested with Matlab version a[copy[frac12]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 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 (2001) [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 (2001) 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. The homogeneous isotropic Boltzmann equation (HIBE) is a fundamental dynamic model for many applications in thermodynamics, econophysics and sociodynamics. Despite recent hardware improvements, the solution of the Boltzmann equation remains extremely challenging from the computational point of view, in particular by deterministic methods (free of stochastic noise). This work aims to improve a deterministic direct method recently proposed [V.V. Aristov, Kluwer Academic Publishers, 2001] for solving the HIBE with a generic collisional kernel and, in particular, for taking care of the late dynamics of the relaxation towards the equilibrium. Essentially (a) the original problem is reformulated in terms of particle kinetic energy (exact particle number and energy conservation during microscopic collisions) and (b) the computation of the relaxation rates is improved by the DVM-like correction, where DVM stands for Discrete Velocity Model (ensuring that the macroscopic conservation laws are exactly satisfied). Both these corrections make possible to derive very accurate reference solutions for this test case. Moreover this work aims to distribute an open-source program (called HOMISBOLTZ), which can be redistributed and/or modified for dealing with different applications, under the terms of the GNU General Public License. The program has been purposely designed in order to be minimal, not only with regards to the reduced number of lines (less than 1000), but also with regards to the coding style (as simple as possible). Program title: HOMISBOLTZ Catalogue identifier: AEGN_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEGN_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License No. of lines in distributed program, including test data, etc.: 23 340 No. of bytes in distributed program, including test data, etc.: 7 635 236 Distribution format: tar.gz 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 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 (2001) [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 (2001) 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. |
| Author | Asinari, Pietro |
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| Cites_doi | 10.1007/s10955-008-9536-9 10.2307/2525289 10.1103/PhysRevE.78.056103 10.1016/j.physa.2009.12.032 |
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| References_xml | – year: 2000 ident: bib015 article-title: Sociodynamics: A Systematic Approach to Mathematical Modelling in the Social Sciences – year: 2001 ident: bib016 article-title: Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows – year: 2001 ident: bib007 article-title: Lattice Boltzmann Equation for Fluid Dynamics and Beyond – volume: vol. LXVI start-page: 88 year: 1872 end-page: 175 ident: bib001 article-title: Weitere Studien über das Wärmegleichgewicht under Gasmolekülen publication-title: Sitzungsberichte der Akademie der Wissenschraften – year: 1980 ident: bib005 article-title: Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas – year: 1995 ident: bib014 article-title: Quantitative Sociodynamics: Stochastic Methods and Models of Social Interaction Processes – volume: 132 start-page: 153 year: 2008 end-page: 170 ident: bib018 article-title: Construction of discrete kinetic models with given invariants publication-title: J. Stat. Phys. – year: 2006 ident: bib003 article-title: Slow Rarefied Flows: Theory and Application to Micro-Electro-Mechanical Systems publication-title: Progress in Mathematical Physics – year: 1998 ident: bib004 article-title: Ludwig Boltzmann: The Man Who Trusted Atoms – volume: 1 start-page: 79 year: 1960 end-page: 106 ident: bib012 article-title: The Pareto–Levy law and the distribution of income publication-title: Internat. Econom. Rev. – year: 2005 ident: bib010 article-title: Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory, Interaction of Mechanics and Mathematics – volume: 36 year: 2005 ident: bib009 article-title: Solution of population balance equations using the direct quadrature method of moments publication-title: J. Aerosol Sci. – year: 1987 ident: bib002 article-title: The Boltzmann Equation and its Applications publication-title: Applied Mathematical Sciences – year: 2002 ident: bib006 article-title: Kinetic Theory and Fluid Dynamics, Modeling and Simulation in Science, Engineering and Technology – year: 2005 ident: bib011 article-title: Econophysics of Wealth Distributions – volume: 389 start-page: 1530 year: 2010 end-page: 1548 ident: bib008 article-title: Factorization symmetry in the lattice Boltzmann method publication-title: Physica A: Statistical Mechanics and its Applications – volume: 78 year: 2008 ident: bib013 article-title: Kinetic equations modelling wealth redistribution: A comparison of approaches publication-title: Phys. Rev. E – year: 1975 ident: bib017 article-title: Théorie Cinétique des Gaz à Répartition Discréte de Vitesses – year: 1998 ident: 10.1016/j.cpc.2010.06.041_bib004 – volume: 36 issue: 43 year: 2005 ident: 10.1016/j.cpc.2010.06.041_bib009 article-title: Solution of population balance equations using the direct quadrature method of moments publication-title: J. Aerosol Sci. – year: 2002 ident: 10.1016/j.cpc.2010.06.041_bib006 – year: 2001 ident: 10.1016/j.cpc.2010.06.041_bib016 – year: 1995 ident: 10.1016/j.cpc.2010.06.041_bib014 – year: 2006 ident: 10.1016/j.cpc.2010.06.041_bib003 article-title: Slow Rarefied Flows: Theory and Application to Micro-Electro-Mechanical Systems – year: 1975 ident: 10.1016/j.cpc.2010.06.041_bib017 – year: 2005 ident: 10.1016/j.cpc.2010.06.041_bib010 – year: 2005 ident: 10.1016/j.cpc.2010.06.041_bib011 – volume: vol. LXVI year: 1872 ident: 10.1016/j.cpc.2010.06.041_bib001_1 article-title: Weitere Studien über das Wärmegleichgewicht under Gasmolekülen – volume: 132 start-page: 153 year: 2008 ident: 10.1016/j.cpc.2010.06.041_bib018 article-title: Construction of discrete kinetic models with given invariants publication-title: J. Stat. Phys. doi: 10.1007/s10955-008-9536-9 – volume: 1 start-page: 79 year: 1960 ident: 10.1016/j.cpc.2010.06.041_bib012 article-title: The Pareto–Levy law and the distribution of income publication-title: Internat. Econom. Rev. doi: 10.2307/2525289 – year: 1987 ident: 10.1016/j.cpc.2010.06.041_bib002 article-title: The Boltzmann Equation and its Applications – volume: vol. II start-page: 88 year: 1966 ident: 10.1016/j.cpc.2010.06.041_bib001_2 – year: 2000 ident: 10.1016/j.cpc.2010.06.041_bib015 – year: 2001 ident: 10.1016/j.cpc.2010.06.041_bib007 – volume: 78 issue: 5 year: 2008 ident: 10.1016/j.cpc.2010.06.041_bib013 article-title: Kinetic equations modelling wealth redistribution: A comparison of approaches publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.78.056103 – year: 1980 ident: 10.1016/j.cpc.2010.06.041_bib005 – volume: 389 start-page: 1530 issue: 8 year: 2010 ident: 10.1016/j.cpc.2010.06.041_bib008 article-title: Factorization symmetry in the lattice Boltzmann method publication-title: Physica A: Statistical Mechanics and its Applications doi: 10.1016/j.physa.2009.12.032 |
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| Title | Nonlinear Boltzmann equation for the homogeneous isotropic case: Minimal deterministic Matlab program |
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