Numerische Mathematik

, Volume 141, Issue 3, pp 627–680 | Cite as

Some examples of kinetic schemes whose diffusion limit is Il’in’s exponential-fitting

  • Laurent GosseEmail author
  • Nicolas Vauchelet


This paper is concerned with diffusive approximations of some numerical schemes for several linear (or weakly nonlinear) kinetic models which are motivated by wide-range applications, including radiative transfer or neutron transport, run-and-tumble models of chemotaxis dynamics, and Vlasov–Fokker–Planck plasma modeling. The well-balanced method applied to such kinetic equations leads to time-marching schemes involving a “scattering S-matrix”, itself derived from a normal modes decomposition of the stationary solution. One common feature these models share is the type of diffusive approximation: their macroscopic densities solve drift-diffusion systems, for which a distinguished numerical scheme is Il’in/Scharfetter–Gummel’s “exponential fitting” discretization. We prove that these well-balanced schemes relax, within a parabolic rescaling, towards such type of discretization by means of an appropriate decomposition of the S-matrix, hence are asymptotic preserving.

Mathematics Subject Classification

65M06 34D15 76M45 76R50 



We gladly thank Prof. Christian Krattenthaller (Vienna) for his kind help in the study of the Haar property satisfied by exponential monomials.


  1. 1.
    Aamodt, R.E., Case, K.M.: Useful identities for half-space problems in linear transport theory. Ann. Phys. 21, 284–301 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ainsworth, M., Dorfler, W.: Fundamental systems of numerical schemes for linear convection–diffusion equations and their relationship to accuracy. Computing 66, 199–229 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Allen, D.N.: A suggested approach to finite difference representation of differential equations. Q. J. Mech. Appl. Math. 15, 11–33 (1962)CrossRefzbMATHGoogle Scholar
  4. 4.
    De Almeida, L.N., Bubba, F., Perthame, B., Pouchol, C.: Energy and implicit discretization of the Fokker–Planck and Keller–Segel type equations. arXiv:1803.10629 (to appear in Networks and Heterogeneous Media)
  5. 5.
    Beals, R.: Partial-range completeness and existence of solutions to two-way diffusion equations. J. Math. Phys. 24, 1932 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Beals, R., Protopopescu, V.: Half-range completeness for the Fokker–Planck equation. J. Stat. Phys. 32, 565–584 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Berger, A.E., Solomon, J.M., Ciment, M.: An analysis of a uniformly accurate difference method for a singular perturbation problem. Math. Comp. 37, 79–94 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bianchini, R., Gosse, L.: A truly two-dimensional discretization of drift-diffusion equations on Cartesian grids. SIAM J. Numer. Anal. 56(5), 2845–2870 (2018).
  9. 9.
    Boscarino, S., Pareschi, L., Russo, G.: Implicit-explicit Runge–Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit. SIAM J. Sci. Comp. 35, 22–51 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brezzi, F., Marini, D., Pietra, P.: Two-dimensional exponential fitting and applications to drift-diffusion models. SIAM J. Numer. Anal. 26, 1342–1355 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Burschka, M.A., Titulaer, U.M.: The kinetic boundary layer for the Fokker–Planck equation with absorbing boundary. J. Stat. Phys. 25(3), 569–582 (1981)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Calvez, V., Raoul, G., Schmeiser, C.: Confinement by biased velocity jumps: aggregation of Escherichia coli. Kineic Relat. Models 8, 651–666 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Calvez, V., Gosse, L., Twarogowska, M.: Traveling chemotactic aggregates at mesoscopic scale and bi-stability. SIAM J. Appl. Math. 77(6), 2224–2249 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cercignani, C.: Mathematical Methods in Kinetic Theory. Plenum, New York (1969)CrossRefzbMATHGoogle Scholar
  15. 15.
    Cercignani, C.: Slow Rarefied Flows. Progress in Mathematical Physics. Theory and Application to Micro-Electro-Mechanical Systems. Birkhäuser, Boston (2006)zbMATHGoogle Scholar
  16. 16.
    Cercignani, C., Sgarra, C.: Half-range completeness for the Fokker–Planck equation with an external force. J. Stat. Phys. 66, 1575–1582 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Chalub, F., Markowich, P., Perthame, B., Schmeiser, C.: Kinetic models for chemotaxis and their drift-diffusion limits. Monats. Math. 142, 123–141 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cheney, E.W.: Introduction to Approximation Theory, 2nd edn. American Mathematical Society, Providence (1998)zbMATHGoogle Scholar
  19. 19.
    Chang, J.S., Cooper, J.: A practical difference scheme for Fokker–Planck equations. J. Comput. Phys. 6, 1–16 (1970)CrossRefzbMATHGoogle Scholar
  20. 20.
    Dolak, Y., Schmeiser, C.: Kinetic models for chemotaxis: hydrodynamic limits and spatio-temporal mechanisms. J. Math. Biol. 51, 595–615 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Emako, C., Tang, M.: Well-balanced and asymptotic preserving schemes for kinetic models. arXiv:1603.03171
  22. 22.
    Fisch, N.J., Kruskal, M.: Separating variables in two-way diffusion equations. J. Math. Phys. 21, 740–750 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gartland Jr., E.C.: On the uniform convergence of the Scharfetter–Gummel discretization in one dimension. SIAM J. Numer. Anal. 30, 749–758 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gasca, M., Micchelli, C. (eds.): Total Positivity and Its Applications, Series Mathematics and Its Applications. Springer, Berlin (1996)Google Scholar
  25. 25.
    Gosse, L.: Asymptotic-preserving and well-balanced scheme for the 1D Cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes. J. Math. Anal. Appl. 388, 964–983 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Gosse, L.: Well-balanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension. Kinetic Relat. Mod. 5, 283–323 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Gosse, L.: Computing Qualitatively Correct Approximations of Balance Laws, vol. 2. Springer, Berlin (2013). ISBN 978-88-470-2891-3zbMATHGoogle Scholar
  28. 28.
    Gosse, L.: Redheffer products and numerical approximation of currents in one-dimensional semiconductor kinetic models. SIAM Multiscale Model. Simul. 12, 1533–1560 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Gosse, L.: A well-balanced scheme able to cope with hydrodynamic limits for linear kinetic models. Appl. Math. Lett. 42, 15–21 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Gosse, L.: A well-balanced and asymptotic-preserving scheme for the one-dimensional linear Dirac equation. BIT Numer. Anal. 55, 433–458 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Gosse, L.: Viscous equations treated with \({{\cal{L}}}\)-splines and Steklov–Poincaré operator in two dimensions. In: Innovative Algorithms and Analysis.
  32. 32.
    Gosse, L.: Aliasing and two-dimensional well-balanced for drift-diffusion equations on square grids, submitted (2018)Google Scholar
  33. 33.
    Gosse, L., Toscani, G.: An asymptotic preserving well-balanced scheme for the hyperbolic heat equation. C.R. Acad. Sci. Paris Série I 334, 1–6 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Gosse, L., Vauchelet, N.: Numerical high-field limits in two-stream kinetic models and 1D aggregation equations. SIAM J. Sci. Comput. 38(1), A412–A434 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Gosse, L., Vauchelet, N.: Hydrodynamic singular regimes in 1+1 kinetic models and spectral numerical methods. J. Math. Anal. Appl. 445(1), 564–603 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Greenberg, J., Alt, W.: Stability results for a diffusion equation with functional shift approximating a chemotaxis model. Trans. Am. Math. Soc. 300, 235–258 (1987)CrossRefzbMATHGoogle Scholar
  37. 37.
    Hershikowitz, D., Rothblum, U.G., Schneider, H.: Classifications of nonnegative matrices using diagonal equivalence. SIAM J. Matrix Anal. Applic. 9, 455–460 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Ilin, A.M.: A difference scheme for a differential equation with a small parameter affecting the highest derivative. Math. Notes Acad. Sci. USSR 6, 237–248 (1969)Google Scholar
  39. 39.
    James, F., Vauchelet, N.: Numerical methods for one-dimensional aggregation equations. SIAM J. Num. Anal. 53(2), 895–916 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Jüngel, A.: Transport Equations for Semiconductors. Lecture Notes in Physics, vol. 773. Springer, Berlin (2009)CrossRefGoogle Scholar
  41. 41.
    Kaper, H.G., Lekkerkerker, C.G., Hejtmanek, J.: Spectral Methods in Linear Transport Theory. Birkhäuser, Basel (1982)zbMATHGoogle Scholar
  42. 42.
    Karlin, S., Studden, W.: Tchebycheff Systems, with Applications in Analysis and Statistics. Wiley, New York (1966)zbMATHGoogle Scholar
  43. 43.
    Krattenthaler, C.: personnal communicationGoogle Scholar
  44. 44.
    Krattenthaler, C.: Watermelon configurations with wall interaction: exact and asymptotic results. J. Phys. Conf. Ser. 42, 179–212 (2006)CrossRefGoogle Scholar
  45. 45.
    Kriese, J.T., Chang, T.S., Siewert, C.E.: Elementary solutions of coupled model equations in the kinetic theory of gases. Int. J. Eng. Sci. 12, 441–470 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Lindström, B.: On the vector representations of induced matroids. Bull. Lond. Math. Soc. 5, 85–90 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, New York (1995)zbMATHGoogle Scholar
  48. 48.
    Nieto, J., Poupaud, F., Soler, J.: High field limit for the Vlasov–Poisson–Fokker–Planck system. Arch. Rat. Mech. Anal. 158, 29–59 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Othmer, H., Hillen, T.: The diffusion limit of transport equations II: Chemotaxis equations. SIAM J. Appl. Math. 62, 1222–1250 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Pareschi, L., Russo, G.: Implicit-explicit Runge–Kutta schemes for stiff systems of differential equations. In: Brugnano , L., Trigiante, D. (eds.) Recent Trends in Numerical Analysis, vol. 3, pp. 269–289 (2000)Google Scholar
  51. 51.
    Pareschi, L., Zanella, M.: Structure preserving schemes for nonlinear Fokker–Planck equations and applications. J. Sci. Comput. 74, 1575–1600 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Poupaud, F., Soler, J.: Parabolic limit and stability of the Vlasov–Poisson–Fokker–Planck system. Math. Models Methods Appl. Sci. 10, 1027–1045 (2001)zbMATHGoogle Scholar
  53. 53.
    Roos, H.-G.: Ten ways to generate the Il’in and related schemes. J. Comput. Appl. Math. 53, 43–59 (1993)CrossRefzbMATHGoogle Scholar
  54. 54.
    Roos, H.-G., Stynes, M., Tobiska, L.: Robust numerical methods for singularly perturbed differential equations. Convection–diffusion-reaction and flow problems; 2nd ed. Springer Series in Computational Mathematics 24 (2008). ISBN: 978-3-540-34466-7Google Scholar
  55. 55.
    Scharfetter, D.L., Gummel, H.K.: Large signal analysis of a silicon read diode oscillator. IEEE Trans. Electron Dev. 16(1), 64–77 (1969)CrossRefGoogle Scholar
  56. 56.
    Sinkhorn, R.: A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Stat. 35, 876–879 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Vein, R., Dale, P.: Determinants and Their Applications in Mathematical Physics. Applied Mathematical Sciences, vol. 134. Springer, Berlin (1999)zbMATHGoogle Scholar
  58. 58.
    Voorhoeve, M.: On the oscillation of exponential polynomials. Math. Zeitschrift 151, 277–294 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Wielonsky, F.: A Rolle’s theorem for real exponential polynomials in the complex domain. J. Math. Pures Appl. 80, 389–408 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Wollman, S., Ozizmir, E.: Numerical approximation of the Vlasov–Poisson–Fokker–Planck system in one dimension. J. Comput. Phys. 202, 602–644 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Istituto per le Applicazioni del CalcoloRomeItaly
  2. 2.Laboratoire Analyse Géométrie et ApplicationsUniversité Paris 13, Sorbonne Paris Cité, CNRS UMR 7539VilletaneuseFrance

Personalised recommendations