High-Order Asymptotic-Preserving Projective Integration Schemes for Kinetic Equations

  • Pauline Lafitte
  • Annelies LejonEmail author
  • Ward Melis
  • Dirk Roose
  • Giovanni Samaey
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 103)


We study a projective integration scheme for a kinetic equation in both the diffusive and hydrodynamic scaling, on which a limiting diffusion or advection equation exists. The scheme first takes a few small steps with a simple, explicit method, such as a spatial centered flux/forward Euler time integration, and subsequently projects the results forward in time over a large, macroscopic time step. With an appropriate choice of the inner step size, the time-step restriction on the outer time step is similar to the stability condition for the limiting equation, whereas the required number of inner steps does not depend on the small-scale parameter. The presented method is asymptotic-preserving, in the sense that the method converges to a standard finite volume scheme for the limiting equation in the limit of vanishing small parameter. We show how to obtain arbitrary-order, general, explicit schemes for kinetic equations as well as for systems of nonlinear hyperbolic conservation laws, and provide numerical results.


Kinetic Equation Projective Integration Collision Kernel Asymptotic Preserve Semiconductor Equation 
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  1. 1.
    D. Aregba-Driollet, R. Natalini, Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37(6), 1973–2004 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    U. Ascher, S. Ruuth, R. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25(2–3), 151–167 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    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(1), A22–A51 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    G. Dimarco, L. Pareschi, Asymptotic preserving implicit-explicit Runge–Kutta methods for nonlinear kinetic equations. SIAM J. Numer. Anal. 51(2), 1064–1087 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    F. Filbet, S. Jin, An asymptotic preserving scheme for the ES-BGK model of the Boltzmann equation. J. Sci. Comput. 46(2), 204–224 (2010)CrossRefMathSciNetGoogle Scholar
  6. 6.
    F. Filbet, S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources. J. Comput. Phys. 229(20), 7625–7648 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    C.W. Gear, I. Kevrekidis, Projective methods for stiff differential equations: problems with gaps in their eigenvalue spectrum. SIAM J. Sci. Comput. 24(4), 1091–1106 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    A. Giuseppe, A. Anile, Moment equations for charged particles: global existence results, in Modeling and Simulation in Science, Engineering and Technology (Birkhäuser, Boston, 2004), pp. 59–80Google Scholar
  9. 9.
    S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21, 1–24 (1999)CrossRefMathSciNetGoogle Scholar
  10. 10.
    P. Lafitte, A. Lejon, G. Samaey, A higher order asymptotic-preserving integration scheme for kinetic equations using projective integration, pp. 1–25.
  11. 11.
    P. Lafitte, W. Melis, G. Samaey, A relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws. (submitted)Google Scholar
  12. 12.
    P. Lafitte, G. Samaey, Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit. SIAM J. Sci. Comput. 34(2), A579–A602 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    B. Sommeijer, Increasing the real stability boundary of explicit methods. Comput. Math. Appl. 19(6), 37–49 (1990)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Pauline Lafitte
    • 1
  • Annelies Lejon
    • 2
    Email author
  • Ward Melis
    • 2
  • Dirk Roose
    • 2
  • Giovanni Samaey
    • 2
  1. 1.Laboratoire de Mathématiques Appliquées aux SystémesEcole Centrale ParisChâtenay-MalabryFrance
  2. 2.Department of Computer ScienceKU LeuvenLeuvenBelgium

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