Abstract
Due to the complexity of hyperbolic and kinetic models discussed in the previous chapters, it is difficult to gain much understanding of the behaviour of the models only from analytical results. As we have already seen throughout this study, numerical approaches are critical when trying to unravel the patterns exhibited by these models. There are a large variety of approaches to discretise and simulate numerically the kinetic and hyperbolic models described in the previous sections. However, due to the intense activity of this field, it is impossible to do a detailed review of all numerical schemes developed over the past 50–60 years. Therefore, in this chapter we briefly discuss some of these approaches, to give the reader a glimpse of the large variety of numerical schemes existent in the literature. We start by discussing a few numerical methods for macroscopic hyperbolic models, followed by a discussion on the numerical methods for more complex (and higher dimension) kinetic equations. We conclude this chapter with a brief overview of different boundary conditions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
G. Dimarco, L. Pareschi, Acta Numer. 23, 369 (2014)
G. Colonna, in Plasma Modelling. Methods and Applications, ed. by G. Colonna, A. D’Angola (IOP, London, 2016), pp. 1–23
R. LeVeque, Numerical Methods for Conservation Laws (Birkhäuser, Basel, 1992)
C.W. Shu, in High-Order Methods for Computational Physics, vol. 9, ed. by T. Barth, H. Deconinck (Springer, Berlin, 1999), pp. 439–582
F. Filbet, G. Russo, in Modelling and Computational Methods for Kinetic Equations, ed. by P. Degod, L. Pareschi, G. Russo (Birkhäuser, Boston, 2004), pp. 117–145
F. Filbet, T. Rey, SIAM J. Sci. Comput. 37(3), A1218 (2015)
L. Pareschi, G. Russo, G. Toscani, Modelling and Numerics of Kinetic Dissipative Systems (Nova Science Publ., New York, 2006)
S. MacNamara, G. Strang, in Splitting Methods in Communication, Imaging, Science and Engineering, ed. by R. Glowinski, S. Osher, W. Yin (Springer, Cham, 2017), pp. 95–114
J. Butcher, Numerical Methods for Ordinary Differential Equations (Wiley, Hoboken, 2008)
W. Hackbusch, Integral Equations: Theory and Numerical Treatment. International Series of Numerical Mathematics (Birkhäuser, Basel, 2012)
W. Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes in C. The Art of Scientific Computing, 3rd edn. (Cambridge University Press, Cambridge, 2007)
R. LeVeque, Finite Volume Methods for Hyperbolic Problems (Cambridge University Press, Cambridge, 2002)
C. Chu, Advances in Applied Mechanics, vol. 18 (Academic, New York, 1979), pp. 285–331
J. Furst, K. Kozel, Math. Bohem. 126(2), 379 (2001)
N. Taniguchi, T. Kobayashi, Comput. Fluids 19(3–4), 287 (1991)
S. Godunov, Math. Sbornik 47, 271 (1959)
A. Harten, J. Comput. Phys. 49(2), 357 (1983)
A. Harten, B. Engquist, S. Osher, S. Chakravarty, J. Comput. Phys. 71(2), 231 (1987)
C.W. Shu, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, ed. by B. Cockburn, C. Johnson, C.W. Shu, E. Tadmor. Lecture Notes in Mathematics, vol. 1697 (Springer, Berlin, 1998), pp. 325–432
C. Shu, SIAM Rev. 51(1), 82 (2009)
E. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics (Springer, Berlin, 2009)
H. Nessyahu, E. Tadmor, J. Comput. Phys. 87, 408 (1990)
P. Roe, J. Comput. Phys. 43, 357 (1981)
B.V. Leer, J. Comput. Phys. 23(3), 263 (1977)
P. Roe, Annu. Rev. Fluid Mech. 18, 337 (1986)
P. Sweby, SIAM J. Numer. Anal. 21(5), 995 (1984)
B.V. Leer, J. Comput. Phys. 14(4), 361 (1974)
G.S. Jiang, E. Tadmor, SIAM J. Sci. Comput. 19(6), 1892 (1998)
E. Godlewski, P. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws (Springer, New York, 1996)
L. Pareschi, G. Russo, ESAIM Proc. 10, 35–75 (2001)
B. Lapeyre, E. Pardoux, R. Sentis, Introduction to Monte-Carlo Methods for Transport and Diffusion Equations (Oxford University Press, Oxford, 2003)
S. Rajasanow, W. Wagner, Stochastic Numerics for the Boltzmann Equation (Springer, Berlin, 2005)
L. Pareschi, G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods (Oxford University Press, Oxford, 2014)
L. Pareschi, G. Toscani, C. Villani, Numer. Math. 93, 527 (2003)
L. Pareschi, B. Perthame, Transp. Theory Stat. Phys. 25(3–5), 369 (1996)
L. Pareschi, G. Russo, Transp. Theory Stat. Phys. 29(3–5), 431 (2000)
L. Pareschi, G. Russo, SIAM J. Numer. Anal. 37, 1217 (2000)
I. Gamba, J. Haack, S. Motsch, J. Comput. Phys. 297, 32 (2015)
P. Degond, L. Pareschi, G. Russo (eds.), Modelling and Computational Methods for Kinetic Equations (Birkhäuser, Boston, 2004)
F. Rogier, J. Schneider, Transp. Theory Stat. Phys. 23(1–3), 313 (1994)
G. Bird, Molecular Gas Dynamics (Oxford University Press, London, 1976)
G. Bird, Molecular Gas Dynamics and Direct Simulation of Gas Flows (Clarendon Press, Oxford, 1994)
L. Pareschi, G. Russo, SIAM J. Sci. Comput. 23(4), 1253 (2001)
K. Nanbu, in Proceedings of the 15th International Symposium on Rarefied Gas Dynamics, ed. by V. Boffi, C. Cercignani (1986), pp. 369–383
H. Babovsky, Math. Methods Appl. Sci. 8, 223 (1986)
I. Boyd, J. Stark, J. Comput. Phys. 80(2), 374 (1989)
L. Pan, G. Liu, B. Khoo, B. Song, J. Micromech. Microeng. 10(1), 21 (2000)
L. Chao, S. Kwak, S. Ansumali, Int. J. Mod. Phys. 25, 1340023 (2014)
G. Dimarco, L. Pareschi, in Hyperbolic Problems: Theory, Numerics, Applications, ed. by S. Benzoni-Gavage, D. Serre (Springer, Berlin, 2008)
L. Pareschi, G. Russo, Transp. Theory Stat. Phys. 29(3–5), 415 (2000)
P. Degond, Panoramas et Syntheses 39–40, 1 (2013)
J. Carrillo, R. Eftimie, F. Hoffmann, Kinetic Relat. Model. 8(3), 413 (2015)
E. Gabetta, L. Pareschi, G. Toscani, SIAM. J. Numer. Anal. 34(6), 2168 (1997)
S. Jin, Riv. Mat. Univ. Parma 3, 177 (2012)
P. Degond, G. Dimarco, L. Mieussens, J. Comput. Phys. 227(2), 1176 (2007)
C. Cercignani, The Boltzmann Equation and Its Applications (Springer, New York, 1987)
G.D.P. Degond, L. Pareschi, Int. J. Numer. Methods Fluids 67(2), 189 (2011)
G. Radtke, N. Hadjiconstantinou, Phys. Rev. E 79, 056711 (2009)
L. Pareschi, R. Caflisch, IMA J. Appl. Math. 135, 57 (2004)
L. Pareschi, ESAIM Proc. 15, 87 (2005)
P. Degond, S. Jin, L. Mieussens, J. Comput. Phys. 209, 665 (2005)
S. Chen, E. Weinan, Y. Liu, C.W. Shu, J. Comput. Phys. 225(2), 1314 (2007)
G. Radtke, J.P. Péraud, N. Hadjiconstantinou, Philos. Trans. R. Soc. A 23, 030606 (2013)
W. Ren, H. Liu, S. Jin, J. Comput. Phys. 276, 380 (2014)
B. Zhang, H. Liu, S. Jin, J. Comput. Phys. 305, 575 (2016)
J. Bourgat, P. LeTallec, B. Berthame, Y. Qiu, Contemp. Math. 157, 377 (1994)
S. Tiwari, J. Comput. Phys. 144(2), 710 (1998)
K. Hadeler, Math. Comput. Model. 31(4–5), 75 (2000)
T. Hillen, Can. Appl. Math. Q. 18(1), 1 (2010)
J. Buhl, D.J.T. Sumpter, I.D. Couzin, J.J. Hale, E. Despland, E.R. Miller, S.J. Simpson, Science 312, 1402 (2006)
A. Portz, A. Seyfried, in Pedestrian and Evacuation Dynamics, ed. by R. Peacock, E. Kuligowski, J. Averill (Springer, Boston, 2011), pp. 577–586
F. Filbert, Multiscale Model. Simul. 10(3), 792 (2012)
J.P. Péraud, C. Landon, N. Hadjiconstantinou, Annu. Rev. Heat Tranf. 17, 205 (2014)
C. Cercignani, Theory and Application of the Boltzmann Equation (Scottish Academic Press, Edinburgh, 1975)
S. Ansumali, I. Karlin, Phys. Rev. E 66(2), 026311 (2002)
C.D. Wilson, R.K. Agarwal, F.G. Tcheremissine, Evaluation of various types of wall boundary conditions for the Boltzmann equation. AIP Conf. Proc. 1333, 146–151 (2011)
S. Jin, L. Pareschi, G. Toscani, SIAM J. Numer. Anal. 38, 312 (2000)
M. Lemou, F. Méhats, SIAM J. Sci. Comput. 34(6), B734 (2012)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Eftimie, R. (2018). Numerical Approaches for Kinetic and Hyperbolic Models. In: Hyperbolic and Kinetic Models for Self-organised Biological Aggregations. Lecture Notes in Mathematics(), vol 2232. Springer, Cham. https://doi.org/10.1007/978-3-030-02586-1_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-02586-1_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-02585-4
Online ISBN: 978-3-030-02586-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)