A Second Order Accurate Kinetic Relaxation Scheme for Inviscid Compressible Flows

  • K. R. ArunEmail author
  • M. Lukáčová-Medvidová
  • Phoolan Prasad
  • S. V. Raghurama Rao
Part of the Notes on Numerical Fluid Mechanics and Multidisciplinary Design book series (NNFM, volume 120)


In this paper we present a kinetic relaxation scheme for the Euler equations of gas dynamics in one space dimension. The method is easily applicable to solve any complex system of conservation laws. The numerical scheme is based on a relaxation approximation for conservation laws viewed as a discrete velocity model of the Boltzmann equation of kinetic theory. The discrete kinetic equation is solved by a splitting method consisting of a convection phase and a collision phase. The convection phase involves only the solution of linear transport equations and the collision phase instantaneously relaxes the distribution function to an equilibrium distribution. We prove that the first order accurate method is conservative, preserves the positivity of mass density and pressure and entropy stable. An anti-diffusive Chapman-Enskog distribution is used to derive a second order accurate method. The results of numerical experiments on some benchmark problems confirm the efficiency and robustness of the proposed scheme.


Boltzmann Equation Euler Equation Kinetic Scheme Order Scheme Relaxation Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aregba-Driollet, D., Natalini, R.: Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37, 1973–2004 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    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)zbMATHCrossRefGoogle Scholar
  3. 3.
    Bouchut, F.: Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Statist. Phys. 95, 113–170 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Cercignani, C.: The Boltzmann equation and its applications. Applied Mathematical Sciences, vol. 67. Springer, New York (1988)zbMATHCrossRefGoogle Scholar
  5. 5.
    Chen, G.Q., Levermore, C.D., Liu, T.-P.: Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. Pure Appl. Math. 47, 787–830 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Deshpande, S.M.: On the Maxwellian distribution, symmetric form, and entropy conservation for the Euler equations. Technical Report 2583, NASA, Langley (1986)Google Scholar
  7. 7.
    Deshpande, S.M.: A second order accurate, kinetic-theory based, method for inviscid compressible flows. Technical Report 2613, NASA, Langley (1986)Google Scholar
  8. 8.
    Deshpande, S.M.: Kinetic flux splitting schemes. In: Hafez, M., Oshima, K. (eds.) Computational Fluid Dynamics Review 1995: a State-of-the-art Reference to the Latest Developments in CFD. Wiley (1995)Google Scholar
  9. 9.
    Einfeldt, B., Munz, C.-D., Roe, P.L., Sjögreen, B.: On Godunov-type methods near low densities. J. Comput. Phys. 92, 273–295 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Estivalezes, J.L., Villedieu, P.: High-order positivity-preserving kinetic schemes for the compressible Euler equations. SIAM J. Numer. Anal. 33, 2050–2067 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Godlewski, E., Raviart, P.-A.: Numerical approximation of hyperbolic systems of conservation laws. Applied Mathematical Sciences, vol. 118. Springer, New York (1996)zbMATHGoogle Scholar
  12. 12.
    Jameson, A., Schmidt, W., Turkel, E.: Numerical solution of the euler equations by finite volume methods using Runge Kutta time stepping schemes. AIAA Paper 81-1259 (1981)Google Scholar
  13. 13.
    Jin, S.: Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 122, 51–67 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Jin, S., Xin, Z.P.: The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm. Pure Appl. Math. 48, 235–276 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Kunik, M., Qamar, S., Warnecke, G.: Second-order accurate kinetic schemes for the ultra-relativistic Euler equations. J. Comput. Phys. 192, 695–726 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Liu, T.-P.: Hyperbolic conservation laws with relaxation. Comm. Math. Phys. 108, 153–175 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Natalini, R.: Recent results on hyperbolic relaxation problems. In: Analysis of Systems of Conservation Laws (Aachen, 1997). Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., vol. 99, pp. 128–198. Chapman & Hall/CRC, Boca Raton (1999)Google Scholar
  18. 18.
    Nessyahu, H., Tadmor, E.: Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Pareschi, L., Russo, G.: Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25, 129–155 (2005)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Perthame, B.: Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal. 27, 1405–1421 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Perthame, B.: Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions. SIAM J. Numer. Anal. 29, 1–19 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Raghurama Rao, S.V., Subba Rao, M.: A simple multidimensional relaxation scheme based on characteristics and interpolation. In: 16th AIAA Computational Fluid Dynamics Conference, Orlando, Florida, June 23-26. American Institute of Aeronautics and Astronautics, AIAA-2003-3535 (2003)Google Scholar
  23. 23.
    Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes II. J. Comput. Phys. 83, 32–78 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Tadmor, E.: Approximate solutions of nonlinear conservation laws. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations (Cetraro, 1997). Lecture Notes in Math., vol. 1697, pp. 1–149. Springer, Berlin (1998)CrossRefGoogle Scholar
  25. 25.
    Toro, E.F.: Riemann solvers and numerical methods for fluid dynamics, 2nd edn. Springer, Berlin (1999); A practical introductionzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • K. R. Arun
    • 1
    Email author
  • M. Lukáčová-Medvidová
    • 2
  • Phoolan Prasad
    • 3
  • S. V. Raghurama Rao
    • 4
  1. 1.Institut für Geometrie und Praktische MathematikRWTH-AachenAachenGermany
  2. 2.Institut für MathematikJohannes Gutenberg Universität MainzMainzGermany
  3. 3.Department of MathematicsIndian Institute of ScienceBangaloreIndia
  4. 4.Department of Aerospace EngineeringIndian Institute of ScienceBangaloreIndia

Personalised recommendations