With cedram.org   version française
Search for an article
Table of contents for this volume | Previous article
Ward Melis; Thomas Rey; Giovanni Samaey
Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations
SMAI-Journal of computational mathematics, 5 (2019), p. 53-88, doi: 10.5802/smai-jcm.43
Article PDF
Class. Math.: 82B40, 76P05, 65M70, 65M08, 65M12
Keywords: Boltzmann equation, BGK equation, Projective Integration, spectral theory, fast spectral scheme

Abstract

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.

Bibliography

[1] R. Alexandre, L. Desvillettes, C. Villani & B. Wennberg, “Entropy dissipation and long-range interactions”, Arch. Ration. Mech. Anal. 152 (2000) no. 4, p. 327-355  MR 1765272
[2] F. Aràndiga, A. Baeza, A. M. Belda & P. Mulet, “Analysis of WENO Schemes for Full and Global Accuracy”, SIAM J. Numer. Anal. 49 (2011) no. 2, p. 893-915  MR 2792400
[3] U. M Ascher, S. J Ruuth & B. TR Wetton, “Implicit-explicit methods for time-dependent partial differential equations”, SIAM J. Numer. Anal. 32 (1995) no. 3, p. 797-823
[4] D. S. Balsara & C.-W. Shu, “Monotonicity Preserving Weighted Essentially Non-oscillatory Schemes with Increasingly High Order of Accuracy”, J. Comput. Phys. 160 (2000) no. 2, p. 405-452
[5] C. Baranger & C. Mouhot, “Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials”, Rev. Mat. Ibér. 21 (2005) no. 3, p. 819-841  MR 2231011
[6] M. Bennoune, M. Lemou & L. Mieussens, “Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics”, J. Comput. Phys. 227 (2008), p. 3781-3803
[7] C. Besse & T. Goudon, “Derivation of a non-local model for diffusion asymptotics—application to radiative transfer problems”, Commun. Comput. Phys 8 (2010) no. 5  MR 2674280
[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. 94 (1954) no. 3
[9] G.A. Bird, Molecular gas dynamics and the direct simulation of gas flows, Oxford University Press, 1994
[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. 35 (2013) no. 1, p. A22-A51
[11] C. Buet & S. Cordier, “An asymptotic preserving scheme for hydrodynamics radiative transfer models”, Numer. Math. 108 (2007) no. 2, p. 199-221
[12] R. E Caflisch, “Monte carlo and quasi-monte carlo methods”, Acta Numer. 7 (1998), p. 1-49
[13] Z. Cai & R. Li, “Numerical Regularized Moment Method of Arbitrary Order for Boltzmann-BGK Equation”, SIAM J. Sci. Comput. 32 (2010) no. 5, p. 2875-2907
[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
[15] T. Carleman, “Sur la théorie de l’équation intégrodifférentielle de Boltzmann”, Acta Math. 60 (1933) no. 1, p. 91-146 Article
[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. 227 (2008) no. 16, p. 7929-7951  MR 2437595
[17] C. Cercignani, The Boltzmann Equation and Its Applications, Springer, 1988
[18] C. Cercignani, R. Illner & M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences 106, Springer-Verlag, New York, 1994  MR 1307620
[19] J.-F. Coulombel, F. Golse & T. Goudon, “Diffusion approximation and entropy-based moment closure for kinetic equations”, Asymptotic Anal. 45 (2005) no. 1, 2, p. 1-39
[20] P. Degond, “Asymptotic-Preserving Schemes for Fluid Models of Plasmas”, Panoramas et Syntheses SMF (2014)
[21] G. Dimarco & L. Pareschi, “Asymptotic preserving implicit-explicit Runge-Kutta methods for nonlinear kinetic equations”, SIAM J. Numer. Anal. 51 (2013) no. 2, p. 1064-1087 Article |  MR 3202241
[22] G. Dimarco & L. Pareschi, “Numerical methods for kinetic equations”, Acta Numer. 23 (2014), p. 369-520  MR 3036998
[23] W. E, B. Engquist, X. Li, W. Ren & E. Vanden-Eijnden, “Heterogeneous multiscale methods: a review”, Commun. Comput. Phys 2 (2007) no. 3, p. 367-450
[24] R.S. Ellis & R.S. Pinsky, “The First and Second Fluid Approximations to the Linearized Boltzmann Equation”, J. Math. Pures Appl. 54 (1975) no. 9, p. 125-156
[25] F. Filbet, “On deterministic approximation of the Boltzmann equation in a bounded domain”, Multiscale Model. Simul. 10 (2012) no. 3, p. 792-817, Preprint  MR 2674294
[26] F. Filbet & S. Jin, “A Class of Asymptotic-Preserving Schemes for Kinetic Equations and Related Problems with Stiff Sources”, J. Comput. Phys. 229 (2010) no. 20, p. 7625-7648  MR 2746671
[27] F. Filbet & C. Mouhot, “Analysis of Spectral Methods for the Homogeneous Boltzmann Equation”, Trans. Amer. Math. Soc. 363 (2011), p. 1947-1980
[28] F. Filbet, C. Mouhot & L. Pareschi, “Solving the Boltzmann Equation in N log2 N”, SIAM J. Sci. Comput. 28 (2007) no. 3, p. 1029-1053
[29] C.W. Gear & I. G. Kevrekidis, “Telescopic projective methods for parabolic differential equations”, J. Comput. Phys. 187 (2003) no. 1, p. 95-109
[30] C.W. Gear & I.G. Kevrekidis, “Projective Methods for Stiff Differential Equations: Problems with Gaps in Their Eigenvalue Spectrum”, SIAM J. Sci. Comput. 24 (2003) no. 4, p. 1091-1106  MR 1977781
[31] G. A. Gerolymos, D. Sénéchal & I. Vallet, “Very-high-order WENO schemes”, J. Comput. Phys. 228 (2009) no. 23, p. 8481-8524
[32] P. Godillon-Lafitte & T. Goudon, “A coupled model for radiative transfer: Doppler effects, equilibrium, and nonequilibrium diffusion asymptotics”, Multiscale Model. Simul. 4 (2005) no. 4, p. 1245-1279
[33] François Golse, The Boltzmann equation and its hydrodynamic limits, in C.n Dafermos, E. Feireisl, ed., Handbook of Differential Equations: Evolutionary Equations Vol. 2, North-Holland, 2005, p. 159–303
[34] L. Gosse & G. Toscani, “Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes”, SIAM J. Numer. Anal. 41 (2003) no. 2, p. 641-658
[35] L. Gosse & G. Toscani, “Asymptotic-preserving & well-balanced schemes for radiative transfer and the Rosseland approximation”, Numer. Math. 98 (2004) no. 2, p. 223-250
[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. 207 (2005), p. 542-567
[37] G.-S. Jiang & C.-W. Shu, “Efficient implementation of weighted WENO schemes”, J. Comput. Phys. 126 (1996), p. 202-228
[38] S. Jin, “Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations”, SIAM J. Sci. Comput. 21 (1999) no. 2, p. 441-454  MR 1718639
[39] S. Jin, “Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review”, Riv. Math. Univ. Parma (N.S.) 3 (2012) no. 2, p. 177-216  MR 2964096
[40] S. Jin, L. Pareschi & G. Toscani, “Uniformly accurate diffusive relaxation schemes for multiscale transport equations”, SIAM J. Numer. Anal. 38 (2000) no. 3, p. 913-936  MR 1781209
[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. 1 (2003) no. 4, p. 715-762  MR 2041455
[42] A. Klar, “An asymptotic preserving numerical scheme for kinetic equations in the low Mach number limit”, SIAM J. Numer. Anal. 36 (1999) no. 5, p. 1507-1527  MR 1694679
[43] A. Klar, “A numerical method for kinetic semiconductor equations in the drift-diffusion limit”, SIAM J. Sci. Comput. 20 (1999) no. 5, p. 1696-1712 (electronic) Article |  MR 1706723
[44] A. Kurganov & S. Tsynkov, “On spectral accuracy of quadrature formulae based on piecewise polynomial interpolations”, IMA J. of Math. Anal. 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. 54 (2016) no. 1, p. 1-33
[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. 340 (2017), p. 1-25
[47] P. Lafitte & G. Samaey, “Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit”, SIAM J. Sci. Comput. 34 (2012) no. 2, p. A579-A602
[48] S. L. Lee & C. W. Gear, “Second-order accurate projective integrators for multiscale problems”, J. Comput. Appl. Math. 201 (2007) no. 1, p. 258-274
[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. 31 (2008) no. 10, p. 334-368
[50] X.-D. Liu, S. Osher & T. Chan, “Weighted essentially non-oscillatory schemes”, J. Comput. Phys. 115 (1994), p. 200-212  MR 1300340
[51] Colin P. M-N., Wladimir L. & J.-C. Passy, “A Well-posed Kelvin-Helmholtz Instability Test and Comparison”, Astrophys. J. Suppl. Ser. 201 (2012) no. 2 Article
[52] W. Melis & G. Samaey, “Telescopic projective integration for kinetic equations with multiple relaxation times”, J. Sci. Comput. 76 (2018), arXiv preprint # 1608.07972  MR 3817804
[53] C. Mouhot & L. Pareschi, “Fast algorithms for computing the Boltzmann collision operator”, Math. Comp. 75 (2006) no. 256, p. 1833-1852 (electronic) Article |  MR 2240637
[54] B. Nicolaenko, Dispersion Laws for Plane Wave Propagation, in F. Grunbaum, ed., The Boltzmann Equation Seminar - 1970 to 1971, Courant Institute of Mathematical Sciences, 1971, p. 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. 37 (2000) no. 4, p. 1217-1245  MR 1756425
[56] R. Rico-Martinez, C. W. Gear & I. G. Kevrekidis, “Coarse projective kMC integration: forward/reverse initial and boundary value problems”, J. Comput. Phys. 196 (2004) no. 2, p. 474-489
[57] L. Saint-Raymond, Hydrodynamic Limits of the Boltzmann Equation, Hydrodynamic Limits of the Boltzmann Equation, Springer, 2009
[58] C.-W. Shu, Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws, Technical report, NASA/CR-97-206253, ICASE Report No. 97-65, 1998  MR 1728856
[59] C.-W. Shu, High Order ENO and WENO Schemes for Computational Fluid Dynamics, in A. Quarteroni, ed., Advanced Numerical Approximations of Nonlinear Hyperbolic Equations, Lect. Notes Comput. Sci. Eng. 9, Springer, 1999, p. 439–582  MR 1712281
[60] Y. Sone, Molecular gas dynamics: theory, techniques, and applications, Springer Science & Business Media, 2007  MR 2274674
[61] H. Struchtrup, Macroscopic transport equations for rarefied gas flows, Interaction of Mechanics and Mathematics, Springer, 2005 Article |  MR 2287368
[62] M. Torrilhon, “Two-dimensional bulk microflow simulations based on regularized Grad’s 13-moment equations”, Mult. Mod. & Sim. 5 (2006) no. 3, p. 695-728
[63] C. Villani, A Review of Mathematical Topics in Collisional Kinetic Theory, Elsevier Science, 2002  MR 1942465
[64] H Von Helmoltz, “Über Discontinuierliche Flüssigkeits-Bewegungen [On the Discontinuous Movements of Fluids]”, Monatsberichte der Königlichen Preussiche Akademie der Wissenschaften zu Berlin 23 (1868) no. 215-228