Projective integration schemes for hyperbolic moment equations

Julian Koellermeier; Giovanni Samaey

doi:10.3934/krm.2021008

Article Contents

2021, Volume 14, Issue 2: 353-387. Doi: 10.3934/krm.2021008

This issue Previous Article Emergent dynamics of a thermodynamic Cucker-Smale ensemble on complete Riemannian manifolds Next Article Mass-conserving weak solutions to the coagulation and collisional breakage equation with singular rates

Projective integration schemes for hyperbolic moment equations

Julian Koellermeier^, and
Giovanni Samaey

Department of Computer Science, NUMA, KU Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium

^* Corresponding author: Julian Koellermeier
^* Corresponding author: Julian Koellermeier

Received: April 30, 2020

Revised: October 31, 2020

Early access: January 2021

Published: April 2021

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Abstract

In this paper, we apply projective integration methods to hyperbolic moment models of the Boltzmann equation and the BGK equation, and investigate the numerical properties of the resulting scheme. Projective integration is an explicit scheme that is tailored to problems with large spectral gaps between slow and (one or many) fast eigenvalue clusters of the model. The spectral analysis of a linearized moment model clearly shows spectral gaps and reveals the multi-scale nature of the model for which projective integration is a matching choice. The combination of the non-intrusive projective integration method with moment models allows for accurate, but efficient simulations with significant speedup, as demonstrated using several 1D and 2D test cases with different collision terms, collision frequencies and relaxation times.

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Mathematics Subject Classification: Primary:76P05, 35Q20, 34E13;Secondary:35B40, 35L02.

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Figure 1. Stability domain (37) of PFE method for $ \delta t = 10^{-4} $, $ \Delta t = 10^{-3} $, $ K = 1 $

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Figure 2. Increasing spectral gap in eigenvalue spectra of HSM4 for constant collision frequency $ \nu = 1 $ and varying $ \tau $ is ideally suited for the application of projective integration

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Figure 3. Additional intermediate cluster in eigenvalue spectra of HSM4 for piecewise constant collision frequency $ \nu \in \{0.1,1\} $ and varying $ \tau $ is ideally suited for the application of a two-level telescopic projective integration

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Figure 4. Extended fast eigenvalue spectra of HSM4 for space-dependent collision frequency $ \nu = \rho(x) \in [1,7] $ and varying $ \tau $ requires a connected stability region of a TPI method

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Figure 5. Shock tube for constant collision frequency $ \nu = 1 $ and varying $ \tau $

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Figure 6. Shock tube for space-dependent collision frequency $ \nu = \rho(x) $ and varying $ \tau $

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Figure 7. Model comparison for shock tube using space-dependent collision frequency $ \nu = \rho(x) $ and varying $ \tau $

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Figure 8. Two-beam test for QBME, constant collision frequency $ \nu = 1 $, and varying $ \tau $

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Figure 9. Two-beam test for QBME, piecewise constant collision frequency $ \nu \in \{0.01,1\} $, third order, and varying $ \tau $

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Figure 10. Computational domain for the forward facing step test case, taken from [41]

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Figure 11. Forward facing step for QBME, BGK collision operator, and varying $ \tau $

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Figure 12. Forward facing step for QBME, Boltzmann collision operator, and varying $ \tau $

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Table 1. Maximum extrapolation factors $ N+K+1 $ for connected stability region depending on $ K $ according to [48]

$ K $	1	2	3	4	5	6	7
$ N+K+1 $	4	6	10.66	13.32	18.21	21.24	26.21
$ N $	2	3	6.66	8.32	12.21	14.24	18.21

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Table 2. Stability of different parameter settings for PFE. QBME model, $ \nu = 1, \tau = 10^{-5} $. Base parameters $ K = 1 $, $ \delta = 1 \cdot 10^{-5} $. Parameters predicted by linear stability analysis indicated by gray column. Instable simulation indicated by red numbers

$ \delta t / 10^{-5} $

$ 1.5 $

$ 1.1 $

$ 1 $

$ 0.9 $

$ 0.5 $

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Table 3. Stability of different parameter settings for TPFE. QBME model, $ \nu = \rho(x), \tau = 10^{-5} $. Each line changes only one parameter from the chosen parameters predicted by linear stability analysis $ K = 6 $, $ \delta_0 = 1.4 \cdot 10^{-6} $, $ \delta_1 = 3 \cdot 10^{-5} $ indicated by gray column. Instable simulation indicated by {red} numbers

$ K $	$ 8 $	$ 7 $	$ 6 $	$ 5 $	$ 4 $
$ \delta t_0 / 10^{-6} $	$ 2.5 $	$ 2 $	$ 1.4 $	$ 1.3 $	$ 1.2 $
$ \delta t_1 / 10^{-5} $	$ 5 $	$ 4 $	$ 3 $	$ 2 $	$ 1 $

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Table 4. Speedup of (T)PI schemes in comparison to standard FE scheme

relaxation time $ \tau $	$ 10^{-2} $	$ 10^{-3} $	$ 10^{-4} $	$ 10^{-5} $	$ 10^{-6} $
shock tube $ \nu = 1 $	1	1	1.925	19.25	192.25
shock tube $ \nu = \rho $	2	1.375	3.93	5.61	8.02
two-beam $ \nu = 1 $	1	1	1.925	19.25	192.25
two-beam $ \nu = \nu_i $	1	1	1.925	9.625	96.25
forward facing step	2	2.5	6.25	15.63	39.06

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