Robust Uncertainty Propagation in Systems of Conservation Laws with the Entropy Closure Method

  • Bruno DesprésEmail author
  • Gaël Poëtte
  • Didier Lucor
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 92)


In this paper, we consider hyperbolic systems of conservation laws subject to uncertainties in the initial conditions and model parameters. In order to solve the underlying uncertain systems, we rely on moment theory and the construction of a moment model in the framework of parametric polynomial approximations. We prove the spectral convergence of this approach for the uncertain inviscid Burgers’ equation. We also emphasize the difficulties arising when applying the standard moment method in the context of uncertain systems of conservation laws. In particular, we focus on two relevant examples: the shallow water equations and the Euler system. Next, we review the entropy-based method inspired by plasma physics and rational extended thermodynamics that we propose in this context. We then study the mathematical structure of the well-posed large systems of discretized partial differential equations arising in this framework. The first aim of this work is the description of some mathematical features of the moment method applied to the modeling of uncertainties in systems of conservation laws. The second objective is to relate theoretical description and understanding to some basic numerical results obtained for the numerical approximation of such uncertain models. All numerical examples come from fluid dynamics inspired problems.


Shallow Water Equation Uncertain System Deterministic System Euler System Entropy Variable 
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.


  1. 1.
    N. Wiener. The Homogeneous Chaos. Amer. J. Math., 60:897–936, 1938.MathSciNetCrossRefGoogle Scholar
  2. 2.
    R.H. Cameron and W.T. Martin. The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals. Annals of Math., 48:385–392, 1947.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    P. D. Lax. Hyperbolic systems of conservation laws and the theory of shock waves. SIAM, 1973. Philadelphia.Google Scholar
  4. 4.
    S. K. Godunov. A Difference Scheme for Numerical Solution of Discontinuous Solution of Hydrodynamic Equations. Math. Sbornik, 47:271–306, 1959. translated US Joint Publ. Res. Service, JPRS 7226, 1969.Google Scholar
  5. 5.
    P. L. Roe. Approximate Riemann solvers, parameter vectors and difference schemes. Journal of Computational Physics, 43:357–372, 1981.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    L. Mathelin, M.Y. Hussaini, and T.A. Zang. Stochastic approaches to uncertainty quantification in CFD simulations. Numer. Algo., 38:209–236, 2005.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Qian-Yong Chen, David Gottlieb, and Jan S. Hesthaven. Uncertainty analysis for the steady-state flows in a dual throat nozzle. Journal of Computational Physics, 204(1):378–398, 2005.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    G. Lin, S.-H. Su, and G.E. Karniadakis. Predicting shock dynamics in the presence of uncertainties. Journal of Computational Physics, 217:260–276, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    G. Lin, C.-H. Su, and G. E. Karniadakis. Random roughness enhances lift in supersonic flow. Phys. Rev. Lett., 99(10):104501, 2007.Google Scholar
  10. 10.
    D. Lucor, C. Enaux, H. Jourdren, and P. Sagaut. Multi-Physics Stochastic Design Optimization: Application to Reacting Flows and Detonation. Comp. Meth. Appl. Mech. Eng., 196:5047–5062, 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    D. Gottlieb and D. Xiu. Galerkin Method for Wave Equations with Uncertain Coefficients. Commun. Comp. Phys., 3:505–518, 2008.MathSciNetzbMATHGoogle Scholar
  12. 12.
    T. Chantrasmi, A. Doostan, and G. Iaccarino. Padé-Legendre approximants for uncertainty analysis with discontinuous response surfaces. Journal of Computational Physics, 228(19):7159–7180, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pettersson P., Iaccarino G., and Nordstrom J. Numerical analysis of the Burgers equation in the presence of uncertainty. Journal of Computational Physics, 228:8394–8412, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    G. Poëtte, B. Després, and D. Lucor. Uncertainty Quantification for Systems of Conservation Laws. J. Comp. Phys., 228(7):2443–2467, 2009.CrossRefzbMATHGoogle Scholar
  15. 15.
    F. Simon, P. Guillen, P. Sagaut, and D. Lucor. A gPC-based approach to uncertain transonic aerodynamics. Compu. Meth. Appl. Mech. Eng., 199:1091–1099, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    J.-C. Chassaing and D. Lucor. Stochastic investigation of flows about airfoils at transonic speeds. AIAA J., 48(5):938–950, 2010.CrossRefGoogle Scholar
  17. 17.
    T.J. Barth. On the propagation of statistical model parameter uncertainty in CFD calculations. Theor. Comput. Dyn., pages 1–28, 2011.Google Scholar
  18. 18.
    G. Poëtte, L. Lucor, and H. Jourdren. A stochastic surrogate model approach applied to calibration of unstable fluid flow experiments. C.R. Acad. Sci. paris, Ser. I, 350(5-6342):319–324, 2012.Google Scholar
  19. 19.
    G. Lin, C.-H. Su, and G. E. Karniadakis. The Stochastic Piston Problem. PNAS, 101(45):15840–15845, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    P.M. Congedo, P. Colonna, C. Corre, J.A.S. Witteveen, and G. Iaccarino. Backward uncertainty propagation method in flow problems: Application to the prediction of rarefaction shock waves. Computer Methods in Applied Mechanics and Engineering, 213-216(0):314–326, 2012.CrossRefGoogle Scholar
  21. 21.
    Ch. Schwab A. Barth and N. Zollinger. Multilevel Monte-Carlo Method for Elliptic PDEs with Stochastic Coefficients. Num. Math., 2011.Google Scholar
  22. 22.
    Ch. Schwab S. Mishra and J. Sukys. Multi-level Monte Carlo finite volume methods for non-linear systems of conservation laws in multi-dimensions. Technical report, ETHZ, 2011.Google Scholar
  23. 23.
    R. Abgrall. A Simple, Flexible and Generic Deterministic Approach to Uncertainty Quantifications in Non Linear Problems: Application to Fluid Flow Problems. Rapport de Recherche INRIA, 2007.Google Scholar
  24. 24.
    J. Tryoen, O. Le Maître, M. Ndjinga, and A. Ern. Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems. Journal of Computational Physics, 229:6485–6511, 1 September 2010. Original Research Article.Google Scholar
  25. 25.
    G. Boillat and T. Ruggeri. Hyperbolic principal subsystems: entropy convexity and subcharacteristic conditions. Arch. Ration. Mech. Anal., 137(4):305–320, 1997.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    G. Chen, C. Levermore, and T. Liu. Hyperbolic Conservation Laws with Stiff Relaxation Terms and Entropy. Comm. Pure Appl. Math., 47:787–830, 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    I. Müller and T. Ruggeri. Rational Extended Thermodynamics, 2nd ed. Springer. Tracts in Natural Philosophy, Volume 37, 1998. Springer-Verlag, New York.Google Scholar
  28. 28.
    Bruno Després. A geometrical approach to nonconservative shocks and elastoplastic shocks. Archive for Rational Mechanics and Analysis, 186(2):275–308(34), 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    C. Dafermos. Hyperbolic conservation laws in continuum physics. Springer Verlag 325, Berlin, 2000.Google Scholar
  30. 30.
    G. Poëtte and D. Lucor. Non Intrusive Iterative Stochastic Spectral Representation with Application to Compressible Gas Dynamics. J. of Comput. Phys., 231:3587–3609, 2012.CrossRefzbMATHGoogle Scholar
  31. 31.
    G. Poëtte, B. Després, and D. Lucor. Treatment of Uncertain Interfaces in Compressible Flows. Comp. Meth. Appl. Math. Engrg., 200:284–308, 2010.CrossRefzbMATHGoogle Scholar
  32. 32.
    K. Veroy, C. Prud’homme, and A. T. Patera. Reduced-basis approximation of the viscous burgers equation: rigorous a posteriori error bounds. Comptes Rendus Mathématique, 337(9):619–624, 2003.Google Scholar
  33. 33.
    D. Xiu and G. E. Karniadakis. The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations. SIAM J. Sci. Comp., 24(2):619–644, 2002.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    X. Wan and G.E. Karniadakis. Beyond Wiener-Askey Expansions: Handling Arbitrary PDFs. SIAM J. Sci. Comp., 27(1–3), 2006.Google Scholar
  35. 35.
    D. Xiu and G.E. Karniadakis. Modeling Uncertainty in Steady State Diffusion Problems via generalized Polynomial Chaos. Comp. Meth. Appl. Mech. Engrg., 191:4927–4948, 2002.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    D. Serre. Systèmes Hyperboliques de Lois de Conservation, partie I. Diderot, 1996. Paris.Google Scholar
  37. 37.
    E.F. Toro. Riemann solver and numerical methods for fluid dynamics. Springer-Verlag, 1997.Google Scholar
  38. 38.
    Michael Junk. Maximum Entropy for Reduced Moment Problems. Math. Mod. Meth. Appl. Sci. Google Scholar
  39. 39.
    Milton Abramowitz and Irene A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, ninth dover printing, tenth gpo printing edition, 1964.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.UMR 7598, Laboratoire Jacques-Louis LionsUniversité Pierre et Marie CurieParis Cedex 05France
  2. 2.CEA, Centre DAM, DIFArpajonFrance
  3. 3.UMR 7190, d’Alembert InstituteUniversité Pierre et Marie CurieParis Cedex 05France

Personalised recommendations