Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations
The SMAI journal of computational mathematics, Volume 5 (2019), pp. 53-88.

We present high-order, fully explicit projective integration schemes for nonlinear collisional kinetic equations such as the BGK and Boltzmann equation. The methods first take a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution. Then, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. The procedure can be recursively repeated on a hierarchy of projective levels to construct telescopic projective integration methods. Based on the spectrum of the linearized collision operator, we deduce that the computational cost of the method is essentially independent of the stiffness of the problem: with an appropriate choice of inner step size, the time step restriction on the outer time step, as well as the number of inner time steps, is independent of the stiffness of the (collisional) source term. In some cases, the number of levels in the telescopic hierarchy depends logarithmically on the stiffness. We illustrate the method with numerical results in one and two spatial dimensions.

Published online:
DOI: 10.5802/smai-jcm.43
Classification: 82B40,  76P05,  65M70,  65M08,  65M12
Keywords: Boltzmann equation, BGK equation, Projective Integration, spectral theory, fast spectral scheme
@article{SMAI-JCM_2019__5__53_0,
author = {Ward Melis and Thomas Rey and Giovanni Samaey},
title = {Projective and telescopic projective integration for the nonlinear {BGK} and {Boltzmann} equations},
journal = {The SMAI journal of computational mathematics},
pages = {53--88},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {5},
year = {2019},
doi = {10.5802/smai-jcm.43},
mrnumber = {3928535},
zbl = {07090179},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.43/}
}
TY  - JOUR
TI  - Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations
JO  - The SMAI journal of computational mathematics
PY  - 2019
DA  - 2019///
SP  - 53
EP  - 88
VL  - 5
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.43/
UR  - https://www.ams.org/mathscinet-getitem?mr=3928535
UR  - https://zbmath.org/?q=an%3A07090179
UR  - https://doi.org/10.5802/smai-jcm.43
DO  - 10.5802/smai-jcm.43
LA  - en
ID  - SMAI-JCM_2019__5__53_0
ER  - 
%0 Journal Article
%T Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations
%J The SMAI journal of computational mathematics
%D 2019
%P 53-88
%V 5
%I Société de Mathématiques Appliquées et Industrielles
%U https://doi.org/10.5802/smai-jcm.43
%R 10.5802/smai-jcm.43
%G en
%F SMAI-JCM_2019__5__53_0
Ward Melis; Thomas Rey; Giovanni Samaey. Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations. The SMAI journal of computational mathematics, Volume 5 (2019), pp. 53-88. doi : 10.5802/smai-jcm.43. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.43/

[1] R. Alexandre; L. Desvillettes; C. Villani; B. Wennberg Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., Volume 152 (2000) no. 4, pp. 327-355 | Article | MR: 1765272 | Zbl: 0968.76076

[2] F. Aràndiga; A. Baeza; A. M. Belda; P. Mulet Analysis of WENO Schemes for Full and Global Accuracy, SIAM J. Numer. Anal., Volume 49 (2011) no. 2, pp. 893-915 | MR: 2792400 | Zbl: 1233.65051

[3] U. M Ascher; S. J Ruuth; B. TR Wetton Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., Volume 32 (1995) no. 3, pp. 797-823 | Article | MR: 1335656 | Zbl: 0841.65081

[4] D. S. Balsara; C.-W. Shu Monotonicity Preserving Weighted Essentially Non-oscillatory Schemes with Increasingly High Order of Accuracy, J. Comput. Phys., Volume 160 (2000) no. 2, pp. 405-452 | MR: 1763821 | Zbl: 0961.65078

[5] C. Baranger; C. Mouhot Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials, Rev. Mat. IbÃ©r., Volume 21 (2005) no. 3, pp. 819-841 | Article | MR: 2231011 | Zbl: 1092.76057

[6] M. Bennoune; M. Lemou; L. Mieussens Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics, J. Comput. Phys., Volume 227 (2008), pp. 3781-3803 | Article | MR: 2403867 | Zbl: 1317.76058

[7] C. Besse; T. Goudon Derivation of a non-local model for diffusion asymptoticsâapplication to radiative transfer problems, Commun. Comput. Phys, Volume 8 (2010) no. 5, 1139 pages | MR: 2674280 | Zbl: 1364.82052

[8] P.L. Bhatnagar; E.P. Gross; M. Krook A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., Volume 94 (1954) no. 3 | Article | Zbl: 0055.23609

[9] G.A. Bird Molecular gas dynamics and the direct simulation of gas flows, Oxford University Press, 1994, 479 pages

[10] S. Boscarino; L. Pareschi; G. Russo Implicit-explicit Runge–Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, SIAM J. Sci. Comput., Volume 35 (2013) no. 1, p. A22-A51 | Article | MR: 3033046 | Zbl: 1264.65150

[11] C. Buet; S. Cordier An asymptotic preserving scheme for hydrodynamics radiative transfer models, Numer. Math., Volume 108 (2007) no. 2, pp. 199-221 | Zbl: 1127.76032

[12] R. E Caflisch Monte carlo and quasi-monte carlo methods, Acta Numer., Volume 7 (1998), pp. 1-49 | Article | MR: 1689431 | Zbl: 0949.65003

[13] Z. Cai; R. Li Numerical Regularized Moment Method of Arbitrary Order for Boltzmann-BGK Equation, SIAM J. Sci. Comput., Volume 32 (2010) no. 5, pp. 2875-2907 | Article | MR: 2729444 | Zbl: 1417.82026

[14] C. Canuto; M.Y. Hussaini; A. Quarteroni; T.A. Zang Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988, xiv+557 pages | Article

[15] T. Carleman Sur la théorie de l’équation intégrodifférentielle de Boltzmann, Acta Math., Volume 60 (1933) no. 1, pp. 91-146 | Article | MR: 1555365 | Zbl: 59.0404.02

[16] J.-A. Carrillo; T. Goudon; P. Lafitte Simulation of fluid and particles flows: Asymptotic preserving schemes for bubbling and flowing regimes, J. Comput. Phys., Volume 227 (2008) no. 16, pp. 7929-7951 | Article | MR: 2437595 | Zbl: 1141.76050

[17] C. Cercignani The Boltzmann Equation and Its Applications, Springer, 1988 | Article | Zbl: 0646.76001

[18] C. Cercignani; R. Illner; M. Pulvirenti The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, Volume 106, Springer-Verlag, New York, 1994, viii+347 pages | MR: 1307620 (96g:82046) | Zbl: 0813.76001

[19] J.-F. Coulombel; F. Golse; T. Goudon Diffusion approximation and entropy-based moment closure for kinetic equations, Asymptotic Anal., Volume 45 (2005) no. 1, 2, pp. 1-39 | MR: 2181257 | Zbl: 1091.35038

[20] P. Degond Asymptotic-Preserving Schemes for Fluid Models of Plasmas, Panoramas et Syntheses, Volume SMF (2014) | Zbl: 1308.76182

[21] G. Dimarco; L. Pareschi Asymptotic preserving implicit-explicit Runge-Kutta methods for nonlinear kinetic equations, SIAM J. Numer. Anal., Volume 51 (2013) no. 2, pp. 1064-1087 | Article | MR: 3036998 | Zbl: 1268.76055

[22] G. Dimarco; L. Pareschi Numerical methods for kinetic equations, Acta Numer., Volume 23 (2014), pp. 369-520 | Article | MR: 3202241 | Zbl: 1398.65260

[23] W. E; B. Engquist; X. Li; W. Ren; E. Vanden-Eijnden Heterogeneous multiscale methods: a review, Commun. Comput. Phys, Volume 2 (2007) no. 3, pp. 367-450 | MR: 2314852 | Zbl: 1164.65496

[24] R.S. Ellis; R.S. Pinsky The First and Second Fluid Approximations to the Linearized Boltzmann Equation, J. Math. Pures Appl., Volume 54 (1975) no. 9, pp. 125-156 | MR: 609540

[25] F. Filbet On deterministic approximation of the Boltzmann equation in a bounded domain, Multiscale Model. Simul., Volume 10 (2012) no. 3, pp. 792-817 (Preprint) | MR: 3022022 | Zbl: 1262.76089

[26] F. Filbet; S. Jin A Class of Asymptotic-Preserving Schemes for Kinetic Equations and Related Problems with Stiff Sources, J. Comput. Phys., Volume 229 (2010) no. 20, pp. 7625-7648 | Article | MR: 2674294 | Zbl: 1202.82066

[27] F. Filbet; C. Mouhot Analysis of Spectral Methods for the Homogeneous Boltzmann Equation, Trans. Amer. Math. Soc., Volume 363 (2011), pp. 1947-1980 | Article | MR: 2746671 | Zbl: 1213.82069

[28] F. Filbet; C. Mouhot; L. Pareschi Solving the Boltzmann Equation in N log2 N, SIAM J. Sci. Comput., Volume 28 (2007) no. 3, pp. 1029-1053 | Article

[29] C.W. Gear; I. G. Kevrekidis Telescopic projective methods for parabolic differential equations, J. Comput. Phys., Volume 187 (2003) no. 1, pp. 95-109 | MR: 1977781 | Zbl: 1018.65116

[30] C.W. Gear; I.G. Kevrekidis Projective Methods for Stiff Differential Equations: Problems with Gaps in Their Eigenvalue Spectrum, SIAM J. Sci. Comput., Volume 24 (2003) no. 4, pp. 1091-1106 | Article | MR: 1976207

[31] G. A. Gerolymos; D. Sénéchal; I. Vallet Very-high-order WENO schemes, J. Comput. Phys., Volume 228 (2009) no. 23, pp. 8481-8524 | Article | MR: 2558763 | Zbl: 1176.65088

[32] P. Godillon-Lafitte; T. Goudon A coupled model for radiative transfer: Doppler effects, equilibrium, and nonequilibrium diffusion asymptotics, Multiscale Model. Simul., Volume 4 (2005) no. 4, pp. 1245-1279 | MR: 2203852 | Zbl: 1236.85006

[33] François Golse The Boltzmann equation and its hydrodynamic limits, Handbook of Differential Equations: Evolutionary Equations Vol. 2 (C.n Dafermos; E. Feireisl, eds.), North-Holland, 2005, pp. 159-303

[34] L. Gosse; G. Toscani Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes, SIAM J. Numer. Anal., Volume 41 (2003) no. 2, pp. 641-658 | Article | MR: 2004192 | Zbl: 1130.82340

[35] L. Gosse; G. Toscani Asymptotic-preserving & well-balanced schemes for radiative transfer and the Rosseland approximation, Numer. Math., Volume 98 (2004) no. 2, pp. 223-250 | MR: 2092741 | Zbl: 1120.65343

[36] A. K. Henrick; T. D. Aslam; J. M. Powers Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points, J. Comput. Phys., Volume 207 (2005), pp. 542-567 | Article | Zbl: 1072.65114

[37] G.-S. Jiang; C.-W. Shu Efficient implementation of weighted WENO schemes, J. Comput. Phys., Volume 126 (1996), pp. 202-228 | Article

[38] S. Jin Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations, SIAM J. Sci. Comput., Volume 21 (1999) no. 2, pp. 441-454 | Article | MR: 1718639 | Zbl: 0947.82008

[39] S. Jin Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, Riv. Math. Univ. Parma (N.S.), Volume 3 (2012) no. 2, pp. 177-216 | MR: 2964096 | Zbl: 1259.82079

[40] S. Jin; L. Pareschi; G. Toscani Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numer. Anal., Volume 38 (2000) no. 3, pp. 913-936 | Article | MR: 1781209 | Zbl: 0976.65091

[41] I. G Kevrekidis; C. W. Gear; J. M. Hyman; P. G. Kevrekidid; O. Runborg; C. Theodoropoulos Equation-free, coarse-grained multiscale computation: Enabling mocroscopic simulators to perform system-level analysis, Commun. Math. Sci., Volume 1 (2003) no. 4, pp. 715-762 | Zbl: 1086.65066

[42] A. Klar An asymptotic preserving numerical scheme for kinetic equations in the low Mach number limit, SIAM J. Numer. Anal., Volume 36 (1999) no. 5, pp. 1507-1527 | Article | MR: 1706723 | Zbl: 0939.76065

[43] A. Klar A numerical method for kinetic semiconductor equations in the drift-diffusion limit, SIAM J. Sci. Comput., Volume 20 (1999) no. 5, p. 1696-1712 (electronic) | Article | MR: 1694679 (2000b:78002) | Zbl: 0930.65100

[44] A. Kurganov; S. Tsynkov On spectral accuracy of quadrature formulae based on piecewise polynomial interpolations, IMA J. of Math. Anal., Volume 25 (2005) no. 4

[45] P. Lafitte; A. Lejon; G. Samaey A high-order asymptotic-preserving scheme for kinetic equations using projective integration, SIAM J. Numer. Anal., Volume 54 (2016) no. 1, pp. 1-33 | MR: 3439764 | Zbl: 1336.65147

[46] P. Lafitte; W. Melis; G. Samaey A high-order relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws, J. Comput. Phys., Volume 340 (2017), pp. 1-25 | Article | MR: 3635826 | Zbl: 1380.65161

[47] P. Lafitte; G. Samaey Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit, SIAM J. Sci. Comput., Volume 34 (2012) no. 2, p. A579-A602 | Article | MR: 2914296 | Zbl: 1252.82068

[48] S. L. Lee; C. W. Gear Second-order accurate projective integrators for multiscale problems, J. Comput. Appl. Math., Volume 201 (2007) no. 1, pp. 258-274 | MR: 2293553 | Zbl: 1110.65063

[49] M. Lemou; L. Mieussens A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput., Volume 31 (2008) no. 10, pp. 334-368 | Article | MR: 2460781 | Zbl: 1187.82110

[50] X.-D. Liu; S. Osher; T. Chan Weighted essentially non-oscillatory schemes, J. Comput. Phys., Volume 115 (1994), pp. 200-212 | Article | MR: 1300340 | Zbl: 0811.65076

[51] Colin P. M-N.; Wladimir L.; J.-C. Passy A Well-posed Kelvin-Helmholtz Instability Test and Comparison, Astrophys. J. Suppl. Ser., Volume 201 (2012) no. 2, 18 pages | Article

[52] W. Melis; G. Samaey Telescopic projective integration for kinetic equations with multiple relaxation times, J. Sci. Comput., Volume 76 (2018), 697â726 pages https://arxiv.org/abs/1608.07972 (arXiv preprint # 1608.07972) | Article | MR: 3817804 | Zbl: 1404.65133

[53] C. Mouhot; L. Pareschi Fast algorithms for computing the Boltzmann collision operator, Math. Comp., Volume 75 (2006) no. 256, p. 1833-1852 (electronic) | Article | MR: 2240637 (2007d:65095) | Zbl: 1105.76043

[54] B. Nicolaenko Dispersion Laws for Plane Wave Propagation, The Boltzmann Equation Seminar - 1970 to 1971 (1971), pp. 125-172

[55] L. Pareschi; G. Russo Numerical Solution of the Boltzmann Equation I : Spectrally Accurate Approximation of the Collision Operator, SIAM J. Numer. Anal., Volume 37 (2000) no. 4, pp. 1217-1245 | Article | MR: 1756425 | Zbl: 1049.76055

[56] R. Rico-Martinez; C. W. Gear; I. G. Kevrekidis Coarse projective kMC integration: forward/reverse initial and boundary value problems, J. Comput. Phys., Volume 196 (2004) no. 2, pp. 474-489 | Zbl: 1053.65005

[57] L. Saint-Raymond Hydrodynamic Limits of the Boltzmann Equation, Hydrodynamic Limits of the Boltzmann Equation, Springer, 2009 no. nÂ° 1971 https://books.google.fr/books?id=ROUILXXb7UUC | Zbl: 1171.82002

[58] C.-W. Shu Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws (1998) (Technical report) | Zbl: 0927.65111

[59] C.-W. Shu High Order ENO and WENO Schemes for Computational Fluid Dynamics, Advanced Numerical Approximations of Nonlinear Hyperbolic Equations (A. Quarteroni, ed.) (Lect. Notes Comput. Sci. Eng.) Volume 9, Springer, Berlin, 1999, pp. 439-582 | MR: 1712281

[60] Y. Sone Molecular gas dynamics: theory, techniques, and applications, Springer Science & Business Media, 2007 | Zbl: 1144.76001

[61] H. Struchtrup Macroscopic transport equations for rarefied gas flows, Interaction of Mechanics and Mathematics, Springer, 2005 | Article | Zbl: 1119.76002

[62] M. Torrilhon Two-dimensional bulk microflow simulations based on regularized Grad’s 13-moment equations, Mult. Mod. & Sim., Volume 5 (2006) no. 3, pp. 695-728 | MR: 2257232 | Zbl: 1388.76340

[63] C. Villani A Review of Mathematical Topics in Collisional Kinetic Theory (Suzanne Friedlander; Denis Serre, eds.), Elsevier Science, 2002, 211 pages

[64] H Von Helmoltz Über Discontinuierliche Flüssigkeits-Bewegungen [On the Discontinuous Movements of Fluids], Monatsberichte der Königlichen Preussiche Akademie der Wissenschaften zu Berlin, Volume 23 (1868) no. 215-228

Cited by Sources: