A Petrov-Galerkin spectral method for the inelastic Boltzmann equation using mapped Chebyshev functions

Jingwei Hu; Jie Shen; Yingwei Wang

doi:10.3934/krm.2020023

Article Contents

2020, Volume 13, Issue 4: 677-702. Doi: 10.3934/krm.2020023

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A Petrov-Galerkin spectral method for the inelastic Boltzmann equation using mapped Chebyshev functions

1.
Department of Mathematics, Purdue University, West Lafayette, IN 47907 USA
2.
SAS Institute, Inc., Cary, NC 27513 USA

^* Corresponding author: Jingwei Hu
^* Corresponding author: Jingwei Hu

Received: June 2019

Revised: January 2020

Published: May 2020

J. Hu's research was supported by NSF grant DMS-1620250 and NSF CAREER grant DMS-1654152. J. Shen's research was supported in part by NSF DMS-1720442 and AFOSR FA9550-16-1-0102

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Abstract

We develop in this paper a Petrov-Galerkin spectral method for the inelastic Boltzmann equation in one dimension. Solutions to such equations typically exhibit heavy tails in the velocity space so that domain truncation or Fourier approximation would suffer from large truncation errors. Our method is based on the mapped Chebyshev functions on unbounded domains, hence requires no domain truncation. Furthermore, the test and trial function spaces are carefully chosen to obtain desired convergence and conservation properties. Through a series of examples, we demonstrate that the proposed method performs better than the Fourier spectral method and yields highly accurate results.

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Mathematics Subject Classification: Primary: 65M70, 35Q20, 41A50; Secondary: 65R20.

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Figure 1. Functions to be approximated

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Figure 2. $ L_2 $ errors in log-log scale

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Figure 3. $ L_2 $ errors in linear-log scale

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Figure 4. Numerical solutions of $ f $ at time $ t = 0, 5, 10, 20 $ with $ e = 0.25 $

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Figure 5. Numerical moments of $ f $ from $ t = 0 $ to $ t = 20 $ with $ e = 0.25 $

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Figure 6. Numerical solutions in the rescaled case at time $ t = 0, 5, 10, 20 $ with $ e = 0.25 $

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Figure 7. Numerical solutions in the rescaled case at time $ t = 0, 5, 10, 20 $ with $ e = 0.75 $

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Figure 8. Errors of numerical moments of $ g $ from $ t = 0 $ to $ t = 20 $

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Figure 9. Numerical solutions of heated case at time $ t = 20 $

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Figure 10. Numerical moments of $ f $ from $ t = 0 $ to $ t = 20 $ with $ e = 0.25, \varepsilon = 0.01 $

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Figure 11. Errors of moments of at $ t = 20 $ with $ e = 0.25, \varepsilon = 0.01 $ with respect to different $ N $

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Figure 12. Numerical moments of $ f $ from $ t = 0 $ to $ t = 20 $ with $ e = 0.75, \varepsilon = 0.01 $

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Figure 13. Errors of moments of at $ t = 20 $ with $ e = 0.75, \varepsilon = 0.01 $ with respect to different $ N $

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