A deterministic-stochastic method for computing the Boltzmann collision integral in $\mathcal{O}(MN)$ operations

Alexander Alekseenko; Truong Nguyen; Aihua Wood

doi:10.3934/krm.2018047

Article Contents

2018, Volume 11, Issue 5: 1211-1234. Doi: 10.3934/krm.2018047 open access

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A deterministic-stochastic method for computing the Boltzmann collision integral in $\mathcal{O}(MN)$ operations

1.
Department of Mathematics, California State University Northridge, Northridge, CA 91330, USA
2.
Department of Mathematics & Statistics, Wright State University, Dayton, OH 45435, USA
3.
Department of Mathematics & Statistics, Air Force Institute of Technology, WPAFB, OH 45433, USA

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

Received: March 2016

Revised: September 2017

Published: May 2018

The first author was supported by NSF grant DMS-1620497 and HPTi PETTT grant PP-SASKY06-001. The third author was supported by AFOSR Grant No. F4FGA04296J00.

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Abstract

We developed and implemented a numerical algorithm for evaluating the Boltzmann collision integral with $O(MN)$ operations, where $N$ is the number of the discrete velocity points and $M <N$. At the base of the algorithm are nodal-discontinuous Galerkin discretizations of the collision operator on uniform grids and a bilinear convolution form of the Galerkin projection of the collision operator. Efficiency of the algorithm is achieved by applying singular value decomposition compression of the discrete collision kernel and by approximating the kinetic solution by a sum of Maxwellian streams using a stochastic likelihood maximization algorithm. Accuracy of the method is established on solutions to the problem of spatially homogeneous relaxation.

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. Magnitudes of eigenvalues of $A_{\alpha\beta}$ for the cases $s_u = s_v = s_w = 1$, $N = 15^3$ (a) and $s = 1$, $N = 33^3$ (b)

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Figure 2. Directional temperatures in the solutions to the problem of spatially homogeneous relaxation for the initial data corresponding to Mach 1.55 (a), (b) and Mach 3.0 (c), (d) shock waves. The solutions shown in Figures (a) and (c) are obtained by using the truncated kernel method for evaluating the collision integral. The solid line corresponds to the solution using the full Boltzmann integral. The new model looses accuracy at about three mean free times for both Mach 1.55 and Mach 3.0 solutions. The solutions shown in Figures (b) and (d) are obtained by using the truncated kernel model for about three mean free times and by using the ellipsoidal-statistical Bhatnagar-Gross-Krook model after

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Table 1. Convergence of errors in the approximation of $\mathcal{A}(f, f)$ by $\hat{\mathcal{A}}_M(f, f)$ for the case $N = 15^3$ with respect to the number $M$ of retained eigenvectors: $\vert\lambda_{M}\vert$ is the magnitude of the $M^{\mathrm{th}}$ eigenvalue; $\varepsilon_1$ is the absolute error between $\mathcal{A}(f, f)$ and $\hat{\mathcal{A}}_M(f, f)$ when $f(t, \vec{x}, \vec{v})$ is a single Maxwellian stream with density $n = 1$, bulk velocity $\bar{\vec{u}} = (0, 0, 0)$ and temperature $T = 0.59$; $\varepsilon_2$ is the relative error between $\mathcal{A}(f, f)$ and $\hat{\mathcal{A}}_M(f, f)$ when $f(t, \vec{x}, \vec{v})$ is a sum of two Maxwellian streams with densities $n_1 = n_2 = 1$, bulk velocities $\bar{\vec{u}}_1 = (0.4, 0, 0)$ and $\bar{\vec{u}}_2 = (-0.4, 0, 0)$ and temperatures $T_1 = T_2 = 0.89$; $\varepsilon_3$ is the relative error between $\mathcal{A}(f, f)$ and $\hat{\mathcal{A}}_M(f, f)$ when $f(t, \vec{x}, \vec{v})$ is a sum of two Maxwellian streams with densities $n_1 = n_2 = 1$, bulk velocities $\bar{\vec{u}}_1 = (0.4, 0, 0)$ and $\bar{\vec{u}}_2 = (-0.4, 0, 0)$ and temperatures $T_1 = T_2 = 0.31$

$M$	$\vert\lambda_{M}\vert$	$\varepsilon_{1}$	$\varepsilon_{2}$	$\varepsilon_{3}$
10	4.1E+0	1.7E-1	3.1E-1	2.1E-1
80	1.0E+0	3.7E-2	5.8E-1	5.3E-2
160	5.7E-1	1.6E-2	2.9E-2	2.0E-2
320	3.6E-1	1.5E-2	2.5E-2	1.8E-2
640	2.2E-1	3.3E-3	8.4E-3	4.4E-3
960	1.1E-1	1.3E-3	5.7E-3	2.2E-3
1120	5.8E-2	2.3E-3	5.9E-3	2.3E-3
1280	4.1E-3	1.7E-5	2.1E-3	3.1E-5
1400	4.6E-5	2.2E-7	2.0E-3	3.9E-7

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Table 2. Convergence of errors in the approximation of $\mathcal{A}(f,f)$ by $\hat{\mathcal{A}}_M(f,f)$ for the case $N=33^3$ with respect to the number $M$ of retained eigenvectors: $\vert\lambda_{M}\vert$ is the magnitude of the $M^{\mathrm{th}}$ eigenvalue; $\varepsilon_1$ is the absolute error between $\mathcal{A}(f,f)$ and $\hat{\mathcal{A}}_M(f,f)$ when $f(t,\vec{x},\vec{v})$ is a single Maxwellian stream with density $n=1$, bulk velocity $\bar{\vec{u}}=(0,0,0)$ and temperature $T=0.89$. $\varepsilon_2$ is the relative error between values of $\mathcal{A}(f,f)$ and values $\hat{\mathcal{A}}_M(f,f)$ when $f(t,\vec{x},\vec{v})$ is a sum of two Maxwellian streams with densities $n_1=n_2=1$, bulk velocities $\bar{\vec{u}}_1=(0.\overline{36},0,0)$ and $\bar{\vec{u}}_2=(-0.\overline{36},0,0)$ and temperatures $T_1=T_2=0.89$. The quantity $\varepsilon_3$ is the relative error between values of $\mathcal{A}(f,f)$ and values $\hat{\mathcal{A}}_M(f,f)$ when $f(t,\vec{x},\vec{v})$ is a sum of two Maxwellian streams with densities $n_1=n_2=1$, bulk velocities $\bar{\vec{u}}_1=(0.\overline{36},0,0)$ and $\bar{\vec{u}}_2=(-0.\overline{36},0,0)$ and temperatures $T_1=T_2=0.31$

$M$	$\vert\lambda_{M}\vert$	$\varepsilon_1$	$\varepsilon_2$	$\varepsilon_3$
10	5.0E+0	1.0E-1	5.2E-1	3.7E-1
100	1.7E+0	4.1E-2	2.0E-1	3.5E-1
500	5.4E-1	5.0E-3	2.3E-2	2.5E-2
1000	2.9E-1	3.3E-3	1.6E-2	1.5E-2
2000	1.7E-1	1.6E-3	7.9E-3	4.7E-3
2900	1.3E-1	1.5E-3	7.5E-3	3.9E-3
4800	9.6E-2	9.2E-4	4.6E-3	2.3E-3
9900	4.3E-2	1.6E-4	3.5E-3	5.3E-4
14400	2.8E-12	2.8E-5	3.5E-3	9.2E-7

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Table 3. CPU time in seconds for computing one time step of the spatially homogeneous solution on a 2.4 GHz processor. Results of simulations for the $N = 15^3$ and $N = 33^3$ grids are presented. $M$ is the number of eigenvectors used in (17). The bottom row gives the time for computing one time step in the method of [3]. Speedup is the ratio of the time used in the new method and the time used by the algorithm of [3]

$n=15$, $s_u=s_v=s_w=1$			$n=33$, $s_u=s_v=s_w=1$
$M$	CPU Time, s	Speedup	$M$	CPU Time, s	Speedup
160	2.0	16.4	4000	444.1	40.0
320	3.8	8.7	6000	668.10	26.6
480	5.4	6.1	8000	890.2	19.9
640	7.0	4.8	10000	1113.4	15.9
960	10.3	3.2	12000	1333.6	13.3
1400	16.0	2.1	14000	1559.4	11.4
DG Boltzmann CPU Time: 33.18 s			DG Boltzmann CPU Time: 17758 s

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Table 4. Results of approximating the solution to the problem of spatially homogeneous relaxation [3] by a sum of two Maxwellian distributions at different stages of the relaxation process. The values of the time are normalized to the mean free time. The initial data for the problem is a sum of two Maxwellians with dimensionless densities, bulk velocities and temperatures, $n_1 = 1.609$, $\bar{\vec{u}}_1 = (0.775, 0, 0)$, $T_1 = 0.3$ and $n_2 = 2.863$, $\bar{\vec{u}}_2 = (0.436, 0, 0)$, $T_2$ = 0.464, respectively. Columns $n_i$, $T_i$, $\bar{u}_i$, $\bar{v}_i$, and $\bar{w}_i$, $i = 1, 2$ show the density, temperature, and the components of the bulk velocity of the approximating Maxwellians, respectively; $\varepsilon_{L^1}$ is the $L^1$ norm error given by (21)

$t/\tau$	$n_1$	$T_1$	$\bar{u}_{1}$	$\bar{v}_{1}$	$\bar{w}_{1}$	$n_2$	$T_2$	$\bar{u}_{2}$	$\bar{v}_{2}$	$\bar{w}_{2}$	$\varepsilon_{L_1}$
0.0	1.54	0.30	0.78	1.1E-2	3.6E-4	2.94	0.46	0.44	-7.4E-3	-2.9E-3	8.5E-3
0.04	1.66	0.30	0.77	1.9E-3	-7.4E-3	2.81	0.47	0.43	-2.5E-4	9.7E-4	8.3E-3
0.45	3.31	0.46	0.48	5.8E-4	1.2E-3	1.16	0.29	0.78	6.3E-3	4.7E-4	7.8E-3
0.87	1.19	0.30	0.76	1.9E-3	1.0E-2	3.28	0.45	0.48	3.3E-4	-2.0E-3	5.8E-3
1.7	1.30	0.33	0.69	6.6E-4	1.5E-2	3.17	0.46	0.50	6.6E-4	-8.9E-3	7.8E-3
2.9	2.93	0.45	0.51	-3.7E-3	7.8E-3	1.54	0.37	0.65	4.5E-3	-9.3E-3	7.1E-3
3.8	1.29	0.36	0.65	-1.6E-2	-1.2E-2	3.18	0.44	0.52	9.0E-3	5.2E-3	6.9E-3
4.2	3.53	0.44	0.53	-4.0E-3	-3.2E-3	0.94	0.36	0.66	1.2E-2	1.3E-2	5.9E-3
4.5	2.21	0.44	0.52	-1.2E-2	-2.7E-2	2.26	0.40	0.60	1.3E-2	2.8E-2	7.0E-3
6.2	2.34	0.42	0.53	2.6E-2	-8.2E-3	2.13	0.42	0.59	-2.7E-2	9.9E-3	6.1E-3
7.9	1.58	0.42	0.53	4.4E-2	-4.3E-2	2.89	0.42	0.57	-2.4E-2	2.3E-2	4.6E-3

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Table 5. Results of approximating the solution to the problem of spatially homogeneous relaxation by a sum of three Maxwellian distributions at different stages of the relaxation process. The initial data for the problem is a sum of two Maxwellians with dimensionless densities, bulk velocities and temperatures, $n_1 = 1.609$, $\bar{\vec{u}}_1 = (0.775, 0, 0)$, $T_1 = 0.3$ and $n_2 = 2.863$, $\bar{\vec{u}}_2 = (0.436, 0, 0)$, $T_2$ = 0.464, respectively

$t/\tau$	$n_1$	$T_1$	$\bar{u}_1$	$\bar{v}_1$	$n_2$	$T_2$	$\bar{u}_2$	$\bar{v}_2$	$n_3$	$T_3$	$\bar{u}_3$	$\bar{v}_3$	$\varepsilon_{L_1}$
0.0	1.35	0.48	0.38	-2.8E-2	1.51	0.44	0.48	3.0E-3	1.61	0.30	0.78	-5.9E-3	6.6E-3
.04	1.97	0.43	0.51	1.0E-2	1.17	0.48	0.38	-1.4E-2	1.33	0.29	0.79	-9.6E-3	6.2E-3
.45	1.86	0.33	0.73	3.6E-3	0.68	0.47	0.45	-4.9E-2	1.93	0.47	0.43	1.3E-2	7.9E-3
.87	0.89	0.46	0.40	-9.0E-2	2.47	0.44	0.53	3.9E-2	1.11	0.31	0.75	-1.1E-2	8.9E-3
1.7	0.92	0.29	0.73	5.5E-3	0.68	0.44	0.52	1.0E-1	2.87	0.45	0.51	-2.7E-2	7.0E-3
2.9	1.59	0.42	0.58	1.4E-2	0.92	0.34	0.67	-1.9E-2	1.96	0.45	0.49	5.7E-2	7.0E-3
3.8	2.01	0.39	0.62	-3.0E-3	1.03	0.43	0.52	5.2E-2	1.43	0.45	0.50	-3.0E-2	6.2E-3
4.2	1.64	0.43	0.54	-2.5E-2	1.03	0.38	0.64	-2.7E-2	1.80	0.43	0.52	-6.7E-3	7.6E-3
4.5	1.50	0.45	0.49	-4.1E-2	0.84	0.37	0.65	-4.0E-2	2.13	0.42	0.57	4.2E-2	5.4E-3
6.2	1.90	0.41	0.56	-2.3E-2	1.81	0.44	0.53	4.3E-2	0.76	0.39	0.63	-3.0E-2	8.0E-3
7.9	1.34	0.42	0.60	2.9E-4	2.27	0.42	0.57	-2.3E-2	0.84	0.41	0.47	6.1E-2	5.9E-3

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