gPC for the Euler Equations

  • Mass Per Pettersson
  • Gianluca Iaccarino
  • Jan Nordström
Part of the Mathematical Engineering book series (MATHENGIN)


We present a stochastic Galerkin formulation of the Euler equations based on the variables introduced by P. L. Roe in 1981 for the well-known and widely used Roe solver. For numerical discretization, a Roe average matrix for the standard MUSCL-Roe scheme with Roe variables is derived. The Roe variable formulation is robust for supersonic problems where failure occurs for the standard implementation based on the conservative variable formulation. The robustness properties can be significantly improved with a stochastic basis that admits an eigenvalue decomposition with constant eigenvectors of the inner triple product matrix that occur frequently in the evaluation of pseudospectral operations. When this is the case, the Roe variables and conservative variables are similar in performance using the same time-step.


Euler Equation Legendre Polynomial Haar Wavelet Eigenvalue Decomposition Conservative Variable 
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  1. 1.
    Debusschere BJ, Najm HN, Pébay PP, Knio OM, Ghanem RG, Le Maître OP (2005) Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J Sci Comput 26:698–719. doi:
  2. 2.
    Le Maître OP, Knio OM (2010) Spectral methods for uncertainty quantification, 1st edn. Springer, Berlin/HeidelbergCrossRefzbMATHGoogle Scholar
  3. 3.
    Le Maître OP, Najm HN, Ghanem RG, Knio OM (2004) Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J Comput Phys 197:502–531. doi: 10.1016/,
  4. 4.
    LeVeque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  5. 5.
    Pettersson P, Abbas Q, Iaccarino G, Nordström J (2009) Efficiency of shock capturing schemes for Burgers’ equation with boundary uncertainty. In: Enumath 2009, the eighth European conference on numerical mathematics and advanced applications, Uppsala, June 29–July 3Google Scholar
  6. 6.
    Pettersson P, Iaccarino G, Nordström J (2014) A stochastic Galerkin method for the Euler equations with Roe variable transformation. J Comput Phys 257, Part A(0):481–500. doi:
  7. 7.
    Poëtte G, Després B, Lucor D (2009) Uncertainty quantification for systems of conservation laws. J Comput Phys 228:2443–2467. doi: 10.1016/,
  8. 8.
    Powell MJD (1970) A Fortran subroutine for solving systems of nonlinear algebraic equations. In: Rabinowitz P (ed) Numerical methods for nonlinear algebraic equations, chap. 7 Gordon and Breach Science Publishers, London/New YorkGoogle Scholar
  9. 9.
    Roache PJ (1988) Verification of codes and calculations. AIAA J 36(5):696–702CrossRefGoogle Scholar
  10. 10.
    Roe PL (1981) Approximate Riemann solvers, parameter vectors, and difference schemes. J Comput Phys 43(2):357–372. doi: 10.1016/0021-9991(81)90128-5,
  11. 11.
    Shunn L, Ham FE, Moin P (2012) Verification of variable-density flow solvers using manufactured solutions. J Comput Phys 231(9):3801–3827CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Tryoen J, Le Maître OP, Ern A (2012) Adaptive anisotropic spectral stochastic methods for uncertain scalar conservation laws. SIAM J Sci Comput 34(5):A2459–A2481CrossRefzbMATHGoogle Scholar
  13. 13.
    Tryoen J, Le Maître OP, Ndjinga M, Ern A (2010) Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems. J Comput Phys 229(18):6485–6511. doi: 10.1016/,
  14. 14.
    Tryoen J, Le Maître OP, Ndjinga M, Ern A (2010) Roe solver with entropy corrector for uncertain hyperbolic systems. J Comput Appl Math 235:491–506. doi:
  15. 15.
    van Leer B (1979) Towards the ultimate conservative difference scheme. V – a second-order sequel to Godunov’s method. J Comput Phys 32:101–136. doi: 10.1016/0021-9991(79)90145-1 CrossRefGoogle Scholar
  16. 16.
    Wan X, Karniadakis GE (2006) Long-term behavior of polynomial chaos in stochastic flow simulations. Comput Methods Appl Math Eng 195:5582–5596CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Mass Per Pettersson
    • 1
  • Gianluca Iaccarino
    • 2
  • Jan Nordström
    • 3
  1. 1.Uni ResearchBergenNorway
  2. 2.Department of Mechanical Engineering and Institute for Computational and Mathematical EngineeringStanford UniversityStanfordUSA
  3. 3.Department of Mathematics Computational MathematicsLinköping UniversityLinköpingSweden

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