Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $ \mathcal{O}(N^2) $ operations using the discrete fourier transform

Alexander Alekseenko; Jeffrey Limbacher

doi:10.3934/krm.2019027

Article Contents

2019, Volume 12, Issue 4: 703-726. Doi: 10.3934/krm.2019027

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Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $ \mathcal{O}(N^2) $ operations using the discrete fourier transform

Alexander Alekseenko^, and
Jeffrey Limbacher

Department of Mathematics, California State University Northridge, Northridge, CA 91330, USA

^* Corresponding author: alexander.alekseenko@csun.edu
^* Corresponding author: alexander.alekseenko@csun.edu

Received: December 31, 2017

Revised: August 31, 2018

Published: May 2019

Authors acknowledge support of NSF grant DMS-1620497. The first author was supported by the AFRL/AFIT MOA Small Grant Program. Computer resources were provided by the Extreme Science and Engineering Discovery Environment, supported by National Science Foundation Grant No. OCI-1053575.

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Abstract

We present a numerical algorithm for evaluating the Boltzmann collision operator with $O(N^2)$ operations based on high order discontinuous Galerkin discretizations in the velocity variable. To formulate the approach, Galerkin projection of the collision operator is written in the form of a bilinear circular convolution. An application of the discrete Fourier transform allows to rewrite the six fold convolution sum as a three fold weighted convolution sum in the frequency space. The new algorithm is implemented and tested in the spatially homogeneous case, and results in a considerable improvement in speed as compared to the direct evaluation. Split and non-split forms of the collision operator are considered, which are forms of the collision operator that have separate and simultaneous evaluations of the gain and loss terms, respectively. Smaller numerical errors are observed in the conserved quantities in simulations using the non-split form.

Keywords:

Kinetic equations,
collision integral,
nodal-discontinuous Galerkin discretizations,
convolution formulation,
dynamics of non-continuum gas.

Mathematics Subject Classification: Primary: 76P05, 76M10; Secondary: 65M60.

Citation:

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Figure 1. Evaluation of the collision operator using split and non-split forms: (a) and (b) the split form evaluated using the Fourier transform; (c) and (d) the split form evaluated directly; (e) and (f) the non-split form evaluated using the Fourier transform

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Figure 2. Relaxation of moments $f_{\varphi_{i, p}} = \int_{R^3} (u_{i}-\bar{u}_{i})^p f(t, \vec{u})\, du$, $i = 1, 2$, $p = 2, 3, 4, 6$ in a mix of Maxwellian streams corresponding to a shock wave with Mach number 3.0 obtained by solving the Boltzmann equation using Fourier and direct evaluations of the collision integral. In the case of $p = 2$, the relaxation of moments is also compared to moments of a DSMC solution [11]

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Figure 3. Relaxation of moments $f_{\varphi_{i, p}}$, $i = 1, 2$, $p = 2, 3, 4, 6$ in a mix of Maxwellian streams corresponding to a shock wave with Mach number 1.55 obtained by solving the Boltzmann equation using Fourier and direct evaluations of the collision integral

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Table 1. CPU times for evaluating the collision operator directly and using the Fourier transform

	DFT		Direct		Speedup
$M$	time, s	$\alpha$	time, s	$\alpha$
9	1.47E-02		1.25E-01		8.5
15	3.94E-01	6.43	4.91E+00	7.18	12.5
21	3.09E+00	6.14	7.80E+01	8.21	25.2
27	1.64E+01	6.65	6.05E+02	8.15	36.7

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Table 2. Absolute errors in conservation of mass and temperature in the discrete collision integral computed using split and non-split formulations

Error in Conservation of Mass					Error in Conservation of Temperature
	Split		Non-split		Split		Non-split
$n$	Fourier	Direct	Fourier	Direct	Fourier	Direct	Fourier	Direct
9	0.37	1.26	1.71E-5	1.92E-5	3.51	1.69	1.71E-2	1.84E-2
15	0.10	1.20	1.45E-5	1.71E-5	0.29	1.25	1.64E-3	3.15E-3
21	0.18	1.18	0.67E-5	0.93E-5	1.38	1.24	5.61E-5	1.75E-3
27	0.18	1.18	0.61E-5	0.86E-5	1.37	1.24	5.40E-4	1.05E-3

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