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 to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
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)
Ascher, U., Ruuth, S., Spiteri, R.: Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25, 151–167 (1997)
Bardos, C., Golse, F., Levermore, D.: Fluid dynamic limits of kinetic equations. I. Formal derivations. J. Stat. Phys. 63, 323–344 (1991)
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)
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)
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)
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)
Caflisch, R.E., Jin, S., Russo, G.: Uniformly accurate schemes for hyperbolic systems with relaxation. SIAM J. Numer. Anal. 34, 246–281 (1997)
Cercignani, C.: The Boltzmann Equation and Its Applications. Springer, New York (1988)
Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-uniform Gases, 3rd edn. Cambridge University Press, Cambridge (1991)
Coron, F., Perthame, B.: Numerical passage from kinetic to fluid equations. SIAM J. Numer. Anal. 28, 26–42 (1991)
Dimarco, G., Pareschi, L.: Asymptotic preserving implicit–explicit Runge–Kutta methods for nonlinear kinetic equations. SIAM J. Numer. Anal. 51, 1064–1087 (2013)
Dimarco, G., Pareschi, L.: Implicit–explicit linear multistep methods for stiff kinetic equations. SIAM J. Numer. Anal. 55, 664–690 (2017)
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)
Filbet, F., Jin, S.: An asymptotic preserving scheme for the ES-BGK model of the Boltzmann equation. J. Sci. Comput. 46, 204–224 (2011)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II: Stiff and Differential-Algebraic Problems. Springer, New York (1987)
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)
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)
Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)
Jin, S.: Runge–Kutta methods for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 122, 51–67 (1995)
Jin, S.: Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review. Riv. Mat. Univ. Parma 3, 177–216 (2012)
Liotta, S.F., Romano, V., Russo, G.: Central schemes for balance laws of relaxation type. SIAM J. Numer. Anal. 38, 1337–1356 (2000)
Pareschi, L., Russo, G.: Implicit–explicit Runge–Kutta methods and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25, 129–155 (2005)
Pieraccini, S., Puppo, G.: Implicit–explicit schemes for BGK kinetic equations. J. Sci. Comput. 1, 1–28 (2007)
Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, Berlin (2013)
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)
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)
Zhang, X.: On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier–Stokes equations. J. Comput. Phys. 328, 301–343 (2017)
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.
About this article
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
- Boltzmann equation
- BGK/ES-BGK models
- IMEX Runge–Kutta schemes
- Compressible Euler equations
- Navier–Stokes equations
Mathematics Subject Classification