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April  2021, 14(2): 353-387. doi: 10.3934/krm.2021008

Projective integration schemes for hyperbolic moment equations

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

* Corresponding author: Julian Koellermeier

Received  May 2020 Revised  November 2020 Published  April 2021 Early access  January 2021

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.

Citation: Julian Koellermeier, Giovanni Samaey. Projective integration schemes for hyperbolic moment equations. Kinetic and Related Models, 2021, 14 (2) : 353-387. doi: 10.3934/krm.2021008
References:
[1]

R. Abgrall and S. Karni, A comment on the computation of non-conservative products, J. Comput. Phys., 229 (2010), 2759-2763.  doi: 10.1016/j.jcp.2009.12.015.

[2]

P. AndriesP. Le TallecJ.-P. Perlat and B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number, Eur. J. Mech. B Fluids, 19 (2000), 813-830.  doi: 10.1016/S0997-7546(00)01103-1.

[3]

K. AokiP. DegondS. Takata and H. Yoshida, Diffusion models for Knudsen compressors, Phys. Fluids, 19 (2007), 117-103.  doi: 10.1063/1.2798748.

[4]

J. D. AuM. Torrilhon and W. Weiss, The shock tube study in extended thermodynamics, Phys. Fluids, 13 (2001), 2423-2432.  doi: 10.1063/1.1381018.

[5]

C. BarangerJ. ClaudelN. Hérouard and L. Mieussens, Locally refined discrete velocity grids for deterministic rarefied flow simulations, AIP Conference Proc., 1501 (2012), 389-396.  doi: 10.1063/1.4769549.

[6]

P. L. BhatnagarE. P. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.  doi: 10.1103/PhysRev.94.511.

[7]

V. V. Bogolepov, Flow past forward-facing small step, J. Appl. Mech. Tech. Phys., 24 (1983), 166-171.  doi: 10.1007/BF00910680.

[8]

R. Bouffanais, Design and Control of Swarm Dynamics, SpringerBriefs in Complexity, Springer, Singapore, 2016. doi: 10.1007/978-981-287-751-2.

[9]

Y. BourgaultD. Broizat and P.-E. Jabin, Convergence rate for the method of moments with linear closure relations, Kinet. Relat. Models, 8 (2015), 1-27.  doi: 10.3934/krm.2015.8.1.

[10]

I. D. Boyd, Predicting breakdown of the continuum equations under rarefied flow conditions, AIP Conference Proc., 663 (2003), 899-906.  doi: 10.1063/1.1581636.

[11]

H. Cabannes, R. Gatignol and L.-S. Luol, The Discrete Boltzmann Equation, Lecture notes, University of California, Berkley, 1980.

[12]

Z. Cai, Numerical simulation of microflows with moment method, in 4th Micro and Nano Flow Conference 2014: Proceedings, Brunel University, 2014.

[13]

Z. CaiY. Fan and R. Li, Globally hyperbolic regularization of Grad's moment system, Comm. Pure Appl. Math., 67 (2014), 464-518.  doi: 10.1002/cpa.21472.

[14]

Z. CaiY. Fan and R. Li, Globally hyperbolic regularization of Grad's moment system in one dimensional space, Commun. Math. Sci., 11 (2013), 547-571.  doi: 10.4310/CMS.2013.v11.n2.a12.

[15]

Z. Cai and M. Torrilhon, Approximation of the linearized Boltzmann collision operator for hard-sphere and inverse-power-law models, J. Comput. Phys., 295 (2015), 617-643.  doi: 10.1016/j.jcp.2015.04.031.

[16]

Z. Cai and M. Torrilhon, On the Holway-Weiss debate: Convergence of the Grad-moment-expansion in kinetic gas theory, Phys. Fluids, 31 (2019). doi: 10.1063/1.5127114.

[17]

A. Canestrelli, Numerical Modelling of Alluvial Rivers by Shock Capturing Methods, Ph.D thesis, Universita' Degli Studi di Padova, 2008.

[18]

A. CanestrelliM. DumbserA. Siviglia and E. F. Toro, Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed, Adv. Water Resources, 33 (2010), 291-303.  doi: 10.1016/j.advwatres.2009.12.006.

[19]

M. CastroJ. M. Gallardo and C. Parés, High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems, Math. Comp., 75 (2006), 1103-1134.  doi: 10.1090/S0025-5718-06-01851-5.

[20]

M. J. CastroP. G. LeFlochM. L. Muñoz-Ruiz and C. Parés, Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes, J. Comput. Phys., 227 (2008), 8107-8129.  doi: 10.1016/j.jcp.2008.05.012.

[21]

M. J. Castro, T. Morales de Luna and C. Parés, Well-balanced schemes and path-conservative numerical methods, in Handbook of Numerical Methods for Hyperbolic Problems, Handb. Numer. Anal., 18, Elsevier/North-Holland, Amsterdam, 2017, 131-175. doi: 10.1016/bs.hna.2016.10.002.

[22]

M. J. Castro, C. Parés, G. Puppo and G. Russo, Central schemes for nonconservative hyperbolic systems, SIAM J. Sci. Comput., 34 (2012), B523-B558. doi: 10.1137/110828873.

[23]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.

[24]

I. CraveroG. PuppoM. Semplice and G. Visconti, CWENO: Uniformly accurate reconstructions for balance laws, Math. Comp., 87 (2018), 1689-1719.  doi: 10.1090/mcom/3273.

[25]

G. Dal MasoP. G. Lefloch and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9), 74 (1995), 483-548. 

[26]

K. DebrabantG. Samaey and P. Zieliński, A micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations, SIAM J. Numer. Anal., 55 (2017), 2745-2786.  doi: 10.1137/16M1066658.

[27]

G. DimarcoR. LoubèreJ. Narski and T. Rey, An efficient numerical method for solving the Boltzmann equation in multidimensions, J. Comput. Phys., 353 (2018), 46-81.  doi: 10.1016/j.jcp.2017.10.010.

[28]

B. Dubroca and L. Mieussens, A conservative and entropic discrete-velocity model for rarefied polyatomic gases, in CEMRACS 1999 (Orsay), ESAIM Proc., 10, Soc. Math. Appl. Indust., Paris, 1999, 127-139. doi: 10.1051/proc: 2001012.

[29]

Y. Fan and J. Koellermeier, Accelerating the convergence of the moment method for the Boltzmann equation using filters, J. Sci. Comput., 84 (2020), 28pp. doi: 10.1007/s10915-020-01251-8.

[30]

Y. FanJ. KoellermeierJ. LiR. Li and M. Torrilhon, Model reduction of kinetic equations by operator projection, J. Stat. Phys., 162 (2016), 457-486.  doi: 10.1007/s10955-015-1384-9.

[31]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.

[32]

P. GrohsR. Hiptmair and S. Pintarelli, Tensor-product discretization for the spatially inhomogeneous and transient Boltzmann equation in two dimensions, SMAI J. Comput. Math., 3 (2017), 219-248.  doi: 10.5802/smai-jcm.26.

[33]

S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review, Riv. Math. Univ. Parma (N.S.), 3 (2012), 177-216. 

[34]

P. Kauf, Multi-Scale Approximation Models for the Boltzmann Equation, Ph.D thesis, ETH Zürich, 2011.

[35]

J. Koellermeier, Derivation and Numerical Solution of Hyperbolic Moment Equations for Rarefied Gas Flows, Dissertation, RWTH Aachen University, Aachen, 2017. doi: 10.18154/RWTH-2017-07475.

[36]

J. Koellermeier and M. J. Castro, High-order non-conservative simulation of hyperbolic moment models, in progress, 2020.

[37]

J. Koellermeier and G. Samaey, Software for: Projective integration schemes for hyperbolic moment equations, Zenodo, 2020. doi: 10.5281/zenodo.3843431.

[38]

J. KoellermeierR. P. Schaerer and M. Torrilhon, A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods, Kinet. Relat. Models, 7 (2014), 531-549.  doi: 10.3934/krm.2014.7.531.

[39]

J. Koellermeier and M. Torrilhon, Hyperbolic moment equations using quadrature-based projection methods, AIP Conference Proc., 1628 (2014), 626-633.  doi: 10.1063/1.4902651.

[40]

J. Koellermeier and M. Torrilhon, Numerical study of partially conservative moment equations in kinetic theory, Commun. Comput. Phys., 21 (2017), 981-1011.  doi: 10.4208/cicp.OA-2016-0053.

[41]

J. Koellermeier and M. Torrilhon, Two-dimensional simulation of rarefied gas flows using quadrature-based moment equations, Multiscale Model. Simul., 16 (2018), 1059-1084.  doi: 10.1137/17M1147548.

[42]

P. LafitteA. Lejon and G. Samaey, A high-order asymptotic-preserving scheme for kinetic equations using projective integration, SIAM J. Numer. Anal., 54 (2016), 1-33.  doi: 10.1137/140966708.

[43]

P. Lafitte and G. Samaey, Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 34 (2012), A579-A602. doi: 10.1137/100795954.

[44]

P. G. LeFloch and A. E. Tzavaras, Existence theory for the Riemann problem for non-conservative hyperbolic systems, C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), 347-352. 

[45]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.

[46]

D. A. LockerbyJ. M. Reese and H. Struchtrup, Switching criteria for hybrid rarefied gas flow solvers, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 1581-1598.  doi: 10.1098/rspa.2008.0497.

[47]

J. McDonald and M. Torrilhon, Affordable robust moment closures for CFD based on the maximum-entropy hierarchy, J. Comput. Phys., 251 (2013), 500-523.  doi: 10.1016/j.jcp.2013.05.046.

[48]

W. MelisT. Rey and G. Samaey, Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations, SMAI J. Comput. Math., 5 (2019), 53-88.  doi: 10.5802/smai-jcm.43.

[49]

W. Melis, T. Rey and G. Samaey, Projective integration for nonlinear BGK kinetic equations, in Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems, Springer Proc. Math. Stat., 200, Springer, Cham, 2017, 145-153. doi: 10.1007/978-3-319-57394-6_16.

[50]

W. Melis and G. Samaey, Telescopic projective integration for linear kinetic equations with multiple relaxation times, J. Sci. Comput., 76 (2018), 697-726.  doi: 10.1007/s10915-017-0635-0.

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R. P. Schaerer and M. Torrilhon, On singular closures for the 5-moment system in kinetic gas theory, Commun. Comput. Phys., 17 (2015), 371-400.  doi: 10.4208/cicp.201213.130814a.

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H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows. Approximation Methods in Kinetic Theory, Interaction of Mechanics and Mathematics, Springer, Berlin, 2005. doi: 10.1007/3-540-32386-4.

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M. Torrilhon, Convergence study of moment approximations for boundary value problems of the Boltzmann-BGK equation, Commun. Comput. Phys., 18 (2015), 529-557.  doi: 10.4208/cicp.061013.160215a.

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M. Torrilhon, Modeling nonequilibrium gas flow based on moment equations, in Annual Review of Fluid Mechanics, Annu. Rev. Fluid Mech., 48, Annual Reviews, Palo Alto, CA, 2016, 429-458. doi: 10.1146/annurev-fluid-122414-034259.

[61]

A. Westerkamp and M. Torrilhon, Slow rarefied gas flow past a cylinder: Analytical solution in comparison to the sphere, AIP Conference Proc., 1501 (2012), 207-214.  doi: 10.1063/1.4769505.

[62]

K. Xu and J.-C. Huang, A unified gas-kinetic scheme for continuum and rarefied flows, J. Comput. Phys., 229 (2010), 7747-7764.  doi: 10.1016/j.jcp.2010.06.032.

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W. ZhaoW.-A. Yong and L.-S. Luo, Stability analysis of a class of globally hyperbolic moment system, Commun. Math. Sci., 15 (2017), 609-633.  doi: 10.4310/CMS.2017.v15.n3.a3.

show all references

References:
[1]

R. Abgrall and S. Karni, A comment on the computation of non-conservative products, J. Comput. Phys., 229 (2010), 2759-2763.  doi: 10.1016/j.jcp.2009.12.015.

[2]

P. AndriesP. Le TallecJ.-P. Perlat and B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number, Eur. J. Mech. B Fluids, 19 (2000), 813-830.  doi: 10.1016/S0997-7546(00)01103-1.

[3]

K. AokiP. DegondS. Takata and H. Yoshida, Diffusion models for Knudsen compressors, Phys. Fluids, 19 (2007), 117-103.  doi: 10.1063/1.2798748.

[4]

J. D. AuM. Torrilhon and W. Weiss, The shock tube study in extended thermodynamics, Phys. Fluids, 13 (2001), 2423-2432.  doi: 10.1063/1.1381018.

[5]

C. BarangerJ. ClaudelN. Hérouard and L. Mieussens, Locally refined discrete velocity grids for deterministic rarefied flow simulations, AIP Conference Proc., 1501 (2012), 389-396.  doi: 10.1063/1.4769549.

[6]

P. L. BhatnagarE. P. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.  doi: 10.1103/PhysRev.94.511.

[7]

V. V. Bogolepov, Flow past forward-facing small step, J. Appl. Mech. Tech. Phys., 24 (1983), 166-171.  doi: 10.1007/BF00910680.

[8]

R. Bouffanais, Design and Control of Swarm Dynamics, SpringerBriefs in Complexity, Springer, Singapore, 2016. doi: 10.1007/978-981-287-751-2.

[9]

Y. BourgaultD. Broizat and P.-E. Jabin, Convergence rate for the method of moments with linear closure relations, Kinet. Relat. Models, 8 (2015), 1-27.  doi: 10.3934/krm.2015.8.1.

[10]

I. D. Boyd, Predicting breakdown of the continuum equations under rarefied flow conditions, AIP Conference Proc., 663 (2003), 899-906.  doi: 10.1063/1.1581636.

[11]

H. Cabannes, R. Gatignol and L.-S. Luol, The Discrete Boltzmann Equation, Lecture notes, University of California, Berkley, 1980.

[12]

Z. Cai, Numerical simulation of microflows with moment method, in 4th Micro and Nano Flow Conference 2014: Proceedings, Brunel University, 2014.

[13]

Z. CaiY. Fan and R. Li, Globally hyperbolic regularization of Grad's moment system, Comm. Pure Appl. Math., 67 (2014), 464-518.  doi: 10.1002/cpa.21472.

[14]

Z. CaiY. Fan and R. Li, Globally hyperbolic regularization of Grad's moment system in one dimensional space, Commun. Math. Sci., 11 (2013), 547-571.  doi: 10.4310/CMS.2013.v11.n2.a12.

[15]

Z. Cai and M. Torrilhon, Approximation of the linearized Boltzmann collision operator for hard-sphere and inverse-power-law models, J. Comput. Phys., 295 (2015), 617-643.  doi: 10.1016/j.jcp.2015.04.031.

[16]

Z. Cai and M. Torrilhon, On the Holway-Weiss debate: Convergence of the Grad-moment-expansion in kinetic gas theory, Phys. Fluids, 31 (2019). doi: 10.1063/1.5127114.

[17]

A. Canestrelli, Numerical Modelling of Alluvial Rivers by Shock Capturing Methods, Ph.D thesis, Universita' Degli Studi di Padova, 2008.

[18]

A. CanestrelliM. DumbserA. Siviglia and E. F. Toro, Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed, Adv. Water Resources, 33 (2010), 291-303.  doi: 10.1016/j.advwatres.2009.12.006.

[19]

M. CastroJ. M. Gallardo and C. Parés, High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems, Math. Comp., 75 (2006), 1103-1134.  doi: 10.1090/S0025-5718-06-01851-5.

[20]

M. J. CastroP. G. LeFlochM. L. Muñoz-Ruiz and C. Parés, Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes, J. Comput. Phys., 227 (2008), 8107-8129.  doi: 10.1016/j.jcp.2008.05.012.

[21]

M. J. Castro, T. Morales de Luna and C. Parés, Well-balanced schemes and path-conservative numerical methods, in Handbook of Numerical Methods for Hyperbolic Problems, Handb. Numer. Anal., 18, Elsevier/North-Holland, Amsterdam, 2017, 131-175. doi: 10.1016/bs.hna.2016.10.002.

[22]

M. J. Castro, C. Parés, G. Puppo and G. Russo, Central schemes for nonconservative hyperbolic systems, SIAM J. Sci. Comput., 34 (2012), B523-B558. doi: 10.1137/110828873.

[23]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.

[24]

I. CraveroG. PuppoM. Semplice and G. Visconti, CWENO: Uniformly accurate reconstructions for balance laws, Math. Comp., 87 (2018), 1689-1719.  doi: 10.1090/mcom/3273.

[25]

G. Dal MasoP. G. Lefloch and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9), 74 (1995), 483-548. 

[26]

K. DebrabantG. Samaey and P. Zieliński, A micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations, SIAM J. Numer. Anal., 55 (2017), 2745-2786.  doi: 10.1137/16M1066658.

[27]

G. DimarcoR. LoubèreJ. Narski and T. Rey, An efficient numerical method for solving the Boltzmann equation in multidimensions, J. Comput. Phys., 353 (2018), 46-81.  doi: 10.1016/j.jcp.2017.10.010.

[28]

B. Dubroca and L. Mieussens, A conservative and entropic discrete-velocity model for rarefied polyatomic gases, in CEMRACS 1999 (Orsay), ESAIM Proc., 10, Soc. Math. Appl. Indust., Paris, 1999, 127-139. doi: 10.1051/proc: 2001012.

[29]

Y. Fan and J. Koellermeier, Accelerating the convergence of the moment method for the Boltzmann equation using filters, J. Sci. Comput., 84 (2020), 28pp. doi: 10.1007/s10915-020-01251-8.

[30]

Y. FanJ. KoellermeierJ. LiR. Li and M. Torrilhon, Model reduction of kinetic equations by operator projection, J. Stat. Phys., 162 (2016), 457-486.  doi: 10.1007/s10955-015-1384-9.

[31]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.

[32]

P. GrohsR. Hiptmair and S. Pintarelli, Tensor-product discretization for the spatially inhomogeneous and transient Boltzmann equation in two dimensions, SMAI J. Comput. Math., 3 (2017), 219-248.  doi: 10.5802/smai-jcm.26.

[33]

S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review, Riv. Math. Univ. Parma (N.S.), 3 (2012), 177-216. 

[34]

P. Kauf, Multi-Scale Approximation Models for the Boltzmann Equation, Ph.D thesis, ETH Zürich, 2011.

[35]

J. Koellermeier, Derivation and Numerical Solution of Hyperbolic Moment Equations for Rarefied Gas Flows, Dissertation, RWTH Aachen University, Aachen, 2017. doi: 10.18154/RWTH-2017-07475.

[36]

J. Koellermeier and M. J. Castro, High-order non-conservative simulation of hyperbolic moment models, in progress, 2020.

[37]

J. Koellermeier and G. Samaey, Software for: Projective integration schemes for hyperbolic moment equations, Zenodo, 2020. doi: 10.5281/zenodo.3843431.

[38]

J. KoellermeierR. P. Schaerer and M. Torrilhon, A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods, Kinet. Relat. Models, 7 (2014), 531-549.  doi: 10.3934/krm.2014.7.531.

[39]

J. Koellermeier and M. Torrilhon, Hyperbolic moment equations using quadrature-based projection methods, AIP Conference Proc., 1628 (2014), 626-633.  doi: 10.1063/1.4902651.

[40]

J. Koellermeier and M. Torrilhon, Numerical study of partially conservative moment equations in kinetic theory, Commun. Comput. Phys., 21 (2017), 981-1011.  doi: 10.4208/cicp.OA-2016-0053.

[41]

J. Koellermeier and M. Torrilhon, Two-dimensional simulation of rarefied gas flows using quadrature-based moment equations, Multiscale Model. Simul., 16 (2018), 1059-1084.  doi: 10.1137/17M1147548.

[42]

P. LafitteA. Lejon and G. Samaey, A high-order asymptotic-preserving scheme for kinetic equations using projective integration, SIAM J. Numer. Anal., 54 (2016), 1-33.  doi: 10.1137/140966708.

[43]

P. Lafitte and G. Samaey, Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 34 (2012), A579-A602. doi: 10.1137/100795954.

[44]

P. G. LeFloch and A. E. Tzavaras, Existence theory for the Riemann problem for non-conservative hyperbolic systems, C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), 347-352. 

[45]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.

[46]

D. A. LockerbyJ. M. Reese and H. Struchtrup, Switching criteria for hybrid rarefied gas flow solvers, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 1581-1598.  doi: 10.1098/rspa.2008.0497.

[47]

J. McDonald and M. Torrilhon, Affordable robust moment closures for CFD based on the maximum-entropy hierarchy, J. Comput. Phys., 251 (2013), 500-523.  doi: 10.1016/j.jcp.2013.05.046.

[48]

W. MelisT. Rey and G. Samaey, Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations, SMAI J. Comput. Math., 5 (2019), 53-88.  doi: 10.5802/smai-jcm.43.

[49]

W. Melis, T. Rey and G. Samaey, Projective integration for nonlinear BGK kinetic equations, in Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems, Springer Proc. Math. Stat., 200, Springer, Cham, 2017, 145-153. doi: 10.1007/978-3-319-57394-6_16.

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Figure 1.  Stability domain (37) of PFE method for $ \delta t = 10^{-4} $, $ \Delta t = 10^{-3} $, $ K = 1 $
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
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
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
Figure 5.  Shock tube for constant collision frequency $ \nu = 1 $ and varying $ \tau $
Figure 6.  Shock tube for space-dependent collision frequency $ \nu = \rho(x) $ and varying $ \tau $
Figure 7.  Model comparison for shock tube using space-dependent collision frequency $ \nu = \rho(x) $ and varying $ \tau $
Figure 8.  Two-beam test for QBME, constant collision frequency $ \nu = 1 $, and varying $ \tau $
Figure 9.  Two-beam test for QBME, piecewise constant collision frequency $ \nu \in \{0.01,1\} $, third order, and varying $ \tau $
Figure 10.  Computational domain for the forward facing step test case, taken from [41]
Figure 11.  Forward facing step for QBME, BGK collision operator, and varying $ \tau $
Figure 12.  Forward facing step for QBME, Boltzmann collision operator, and varying $ \tau $
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
$ 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
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 $
$ \delta t / 10^{-5} $ $ 1.5 $ $ 1.1 $ $ 1 $ $ 0.9 $ $ 0.5 $
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 $
$ 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 $
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
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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