Entropy Production by Implicit Runge–Kutta Schemes

  • Carlos LozanoEmail author


This paper follows up on the author’s recent paper “Entropy Production by Explicit Runge–Kutta schemes” (Lozano in J Sci Comput 76(1):521–564, 2018., where a formula for the production of entropy by fully discrete schemes with explicit Runge–Kutta time integrators was presented. In this paper, the focus is on implicit Runge–Kutta schemes, for which the fully discrete numerical entropy evolution scheme is derived and tested.


Implicit Runge–Kutta-schemes Entropy production Entropy stability 



This work has been supported by the Spanish Ministry of Defence and INTA under the research program “Termofluidodinámica” (IGB99001).


  1. 1.
    Lozano, C.: Entropy production by explicit Runge–Kutta Schemes. J. Sci. Comput. 76(1), 521–564 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Butcher, J.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, Chichester (2008)CrossRefzbMATHGoogle Scholar
  3. 3.
    Kennedy, C.A., Carpenter, M.H.: Diagonally implicit Runge–Kutta methods for ordinary differential equations. A review. NASA/TM–2016–219173 (2016)Google Scholar
  4. 4.
    Gottlieb, S., Ketcheson, D.: Time discretization techniques, chapter 21. In: Abgrall, R., Shu, C.W. (eds.) Handbook of numerical methods for hyperbolic problems, vol. 17, pp. 549–583. Elsevier, Amsterdam (2016)Google Scholar
  5. 5.
    Persson, P.-O., Willis, D., Peraire, J.: The numerical simulation of flapping wings at low Reynolds numbers. In: AIAA Paper 2010-724 (2010)Google Scholar
  6. 6.
    Bijl, H., Carpenter, M.H., Vatsa, V.N., Kennedy, C.A.: Implicit time integration schemes for the unsteady compressible Navier-Stokes equations: laminar flow. J. Comput. Phys. 179(1), 313–329 (2002). CrossRefzbMATHGoogle Scholar
  7. 7.
    Jameson, A.: Evaluation of fully implicit Runge Kutta schemes for unsteady flow calculations. J. Sci. Comput. 73(2–3), 819–852 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lax, P.: Shock waves and entropy. In: Zarantonello, E. (ed.) Contributions to Nonlinear Functional Analysis, pp. 603–634. Academic Press, New York (1971)CrossRefGoogle Scholar
  9. 9.
    Harten, A., Hyman, J.M., Lax, P.D., Keyfitz, B.: On finite-difference approximations and entropy conditions for shocks. Commun. Pure Appl. Math. 29, 297–322 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Osher, S.: Riemann solvers, the entropy condition, and difference approximations. SIAM J. Numer. Anal. 21(2), 217–235 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Tadmor, E.: Entropy stable schemes, chapter 18. In: Abgrall, R., Shu, C.W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems, vol. 17, pp. 467–493. Elsevier, Amsterdam (2016)Google Scholar
  12. 12.
    Tadmor, E.: Entropy Stability Theory for Difference Approximations of Nonlinear Conservation Laws and Related Time-dependent Problems. Acta Numerica 12, 451–512 (2003). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lefloch, P.G., Mercier, J.M., Rohde, C.: Fully discrete, entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal. 40(5), 1968–1992 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fjordholm, U., Mishra, S., Tadmor, E.: Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal. 50(2), 544–573 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jiang, G.-S., Shu, C.-W.: On a cell entropy inequality for discontinuous Galerkin method. Math. Comp. 62, 531–538 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Shu, C.W.: Discontinuous Galerkin methods: General approach and stability. In: Numerical Solutions of Partial Differential Equations, Advanced Courses in Mathematics CRM Barcelona, pp. 149–201 (2009)Google Scholar
  17. 17.
    Hiltebrand, A., Mishra, S.: Entropy stable shock capturing streamline diffusion space-time discontinuous Galerkin (DG) methods for systems of conservation laws. Numer. Math. 126, 103–151 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Qiu, J., Zhang, Q.: Stability, error estimate and limiters of discontinuous Galerkin methods. In: Abgrall, R., Shu, C.W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems, chapter 7, vol. 17, pp. 147–171. Elsevier, Amsterdam (2016)Google Scholar
  19. 19.
    Chen, T., Shu, C.-W.: Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. J. Comput. Phys. 345, 427–461 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Carpenter, M., Fisher, T., Nielsen, E., Parsan, M., Svärd, M., Yamaleev, N.: Entropy stable summation-by-parts formulations for compressible computational fluid dynamics, chapter 19. In: Abgrall, R., Shu, C.W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems, vol. 17, pp. 495–524. Elsevier, Amsterdam (2016)Google Scholar
  21. 21.
    Jameson, A.: The construction of discretely conservative finite volume schemes that also globally conserve energy or entropy. J. Sci. Comput. 34, 152–187 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jameson, A.: Formulation of kinetic energy preserving conservative schemes for gas dynamics and direct numerical simulation of one-dimensional viscous compressible flow in a shock tube using entropy and kinetic energy preserving schemes. J. Sci. Comput. 34, 188–208 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ismail, F., Roe, P.L.: Affordable, entropy-consistent Euler flux functions II: entropy production at shocks. J. Comput. Phys. 228(15), 5410–5436 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Diosdady, L., Murman, S.: Higher-order methods for compressible turbulent flows using entropy variables. In: AIAA Paper 2015-0294, 53rd AIAA Aerospace Sciences Meeting, Kissimmee, FL, Jan 5, 2015Google Scholar
  25. 25.
    Gouasmi, A., Murman, S., Duraisamy, K.: Entropy conservative schemes and the receding flow problem. Preprint arXiv:1801.10132v1 [math.NA], January 30, 2018
  26. 26.
    Gouasmi, A., Duraisamy, K., Murman, S.: On entropy stable temporal fluxes. Preprint arXiv:1807.03483v2 [math.NA], July 21, 2018
  27. 27.
    Fjordholm, U., Mishra, S., Tadmor, E.: Energy preserving and energy stable schemes for the shallow water equations. In: Cucker, F., Pinkus, A., Todd, M. (eds.), Proceedings of Foundations of Computational Mathematics, London Math. Soc. Lecture Notes Ser. 36393-139, Hong Kong (2009)Google Scholar
  28. 28.
    Merriam, M.L.: An entropy-based approach to nonlinear stability. NASA TM-101086, March 1989Google Scholar
  29. 29.
    Harten, A.: On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49(1), 151–164 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tadmor, E.: Entropy functions for symmetric systems of conservation laws. J. Math. Anal. Appl. 122(2), 355–359 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Burrage, K.: Stability and efficiency properties of implicit Runge–Kutta methods. PhD Thesis, University of Auckland, 1978Google Scholar
  32. 32.
    Lambert, J.: Numerical methods for ordinary differential systems. The initial value problem. Wiley, ISBN 0-471-92990-5, 1991Google Scholar
  33. 33.
    Burrage, K., Butcher, J.: Stability criteria for implicit Runge–Kutta methods. SIAM J. Numer. Anal. 16, 46–57 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kraaijevanger, J., Spijker, M.N.: Algebraic stability and error propagation in Runge–Kutta methods. Appl. Numer. Math. 5(1–2), 71–87 (1989). MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Zakerzadeh, H., Fjordholm, U.: High-order accurate, fully discrete entropy stable schemes for scalar conservation laws. IMA J. Numer. Anal. 36(2), 633–654 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Tadmor, E., Zhong, W.: Novel entropy stable schemes for 1D and 2D fluid equations. In: Benzoni-Gavage, S., Serre, D. (eds.) Hyperbolic problems: theory, numerics, applications, pp. 1111–1119. Springer, Berlin (2008)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Computational Aerodynamics GroupNational Institute for Aerospace Technology (INTA)Torrejón de ArdozSpain

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