Skip to main content
Springer Nature Link
Log in

On a Class of Implicit–Explicit Runge–Kutta Schemes for Stiff Kinetic Equations Preserving the Navier–Stokes Limit

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Implicit–explicit (IMEX) Runge–Kutta (RK) schemes are popular high order time discretization methods for solving stiff kinetic equations. As opposed to the compressible Euler limit (leading order asymptotics of the Boltzmann equation as the Knudsen number \(\varepsilon \) goes to zero), their asymptotic behavior at the Navier–Stokes (NS) level (next order asymptotics) was rarely studied. In this paper, we analyze a class of existing IMEX RK schemes and show that, under suitable initial conditions, they can capture the NS limit without resolving the small parameter \(\varepsilon \), i.e., \(\varepsilon =o(\Delta t)\), \(\Delta t^m=o(\varepsilon )\), where m is the order of the explicit RK part in the IMEX scheme. Extensive numerical tests for BGK and ES-BGK models are performed to verify our theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Andries, P., Le Tallec, P., Perlat, J.-P., Perthame, B.: The Gaussian-BGK model of Boltzmann equation with small Prandtl number. Eur. J. Mech. B Fluids 19, 813–830 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ascher, U., Ruuth, S., Spiteri, R.: Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25, 151–167 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bardos, C., Golse, F., Levermore, D.: Fluid dynamic limits of kinetic equations. I. Formal derivations. J. Stat. Phys. 63, 323–344 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bennoune, M., Lemou, M., Mieussens, L.: Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier–Stokes asymptotics. J. Comput. Phys. 227, 3781–3803 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525 (1954)

    Article  MATH  Google Scholar 

  6. Boscarino, S., Pareschi, L.: On the asymptotic properties of IMEX Runge–Kutta schemes for hyperbolic balance laws. J. Comput. Appl. Math. 316, 60–73 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  7. Boscarino, S., Pareschi, L., Russo, G.: Implicit–explicit Runge–Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit. SIAM J. Sci. Comput. 35, A22–A51 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Caflisch, R.E., Jin, S., Russo, G.: Uniformly accurate schemes for hyperbolic systems with relaxation. SIAM J. Numer. Anal. 34, 246–281 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cercignani, C.: The Boltzmann Equation and Its Applications. Springer, New York (1988)

    Book  MATH  Google Scholar 

  10. Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-uniform Gases, 3rd edn. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

  11. Coron, F., Perthame, B.: Numerical passage from kinetic to fluid equations. SIAM J. Numer. Anal. 28, 26–42 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dimarco, G., Pareschi, L.: Asymptotic preserving implicit–explicit Runge–Kutta methods for nonlinear kinetic equations. SIAM J. Numer. Anal. 51, 1064–1087 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dimarco, G., Pareschi, L.: Implicit–explicit linear multistep methods for stiff kinetic equations. SIAM J. Numer. Anal. 55, 664–690 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  14. Filbet, F., Jin, S.: A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources. J. Comput. Phys. 229, 7625–7648 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Filbet, F., Jin, S.: An asymptotic preserving scheme for the ES-BGK model of the Boltzmann equation. J. Sci. Comput. 46, 204–224 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II: Stiff and Differential-Algebraic Problems. Springer, New York (1987)

    Book  MATH  Google Scholar 

  17. Holway, L.: Kinetic theory of shock structure using an ellipsoidal distribution function. In: Proceedings of the 4th International Symposium on Rarefied Gas Dynamics, vol. I, pp. 193–215. Academic Press, New York (1966)

  18. Hu, J., Jin, S., Li, Q.: Asymptotic-preserving schemes for multiscale hyperbolic and kinetic equations, chapter 5. In: Abgrall, R., Shu, C.-W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems, pp. 103–129. North-Holland, Amsterdam (2017)

    Chapter  Google Scholar 

  19. Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  20. Jin, S.: Runge–Kutta methods for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 122, 51–67 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  21. Jin, S.: Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review. Riv. Mat. Univ. Parma 3, 177–216 (2012)

    MATH  MathSciNet  Google Scholar 

  22. Liotta, S.F., Romano, V., Russo, G.: Central schemes for balance laws of relaxation type. SIAM J. Numer. Anal. 38, 1337–1356 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Pareschi, L., Russo, G.: Implicit–explicit Runge–Kutta methods and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25, 129–155 (2005)

    MATH  MathSciNet  Google Scholar 

  24. Pieraccini, S., Puppo, G.: Implicit–explicit schemes for BGK kinetic equations. J. Sci. Comput. 1, 1–28 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, Berlin (2013)

    Google Scholar 

  26. Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Friedlander, S., Serre, D. (eds.) Handbook of Mathematical Fluid Mechanics, vol. I, pp. 71–305. North-Holland, Amsterdam (2002)

    Google Scholar 

  27. Xiong, T., Jang, J., Li, F., Qiu, J.-M.: High order asymptotic preserving nodal discontinuous Galerkin IMEX schemes for the BGK equation. J. Comput. Phys. 284, 70–94 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  28. Zhang, X.: On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier–Stokes equations. J. Comput. Phys. 328, 301–343 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Jingwei Hu & Xiangxiong Zhang

Authors
  1. Jingwei Hu
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Xiangxiong Zhang
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Jingwei Hu.

Additional information

J. Hu’s research was supported by NSF Grant DMS-1620250 and NSF CAREER Grant DMS-1654152. Support from DMS-1107291: RNMS KI-Net is also gratefully acknowledged. X. Zhang’s research was supported by NSF Grant DMS-1522593.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, J., Zhang, X. On a Class of Implicit–Explicit Runge–Kutta Schemes for Stiff Kinetic Equations Preserving the Navier–Stokes Limit. J Sci Comput 73, 797–818 (2017). https://doi.org/10.1007/s10915-017-0499-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0499-3

Keywords

Mathematics Subject Classification