Essentially Non-oscillatory Stencil Selection and Subcell Resolution in Uncertainty Quantification

  • Jeroen A. S. WitteveenEmail author
  • Gianluca Iaccarino
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 92)


The essentially non-oscillatory stencil selection and subcell resolution robustness concepts from finite volume methods for computational fluid dynamics are extended to uncertainty quantification for the reliable approximation of discontinuities in stochastic computational problems. These two robustness principles are introduced into the simplex stochastic collocation uncertainty quantification method, which discretizes the probability space using a simplex tessellation of sampling points and piecewise higher-degree polynomial interpolation. The essentially non-oscillatory stencil selection obtains a sharper refinement of discontinuities by choosing the interpolation stencil with the highest polynomial degree from a set of candidate stencils for constructing the local response surface approximation. The subcell resolution approach achieves a genuinely discontinuous representation of random spatial discontinuities in the interior of the simplexes by resolving the discontinuity location in the probability space explicitly and by extending the stochastic response surface approximations up to the predicted discontinuity location. The advantages of the presented approaches are illustrated by the results for a step function, the linear advection equation, a shock tube Riemann problem, and the transonic flow over the RAE 2822 airfoil.


Monte Carlo Transonic Flow Response Surface Approximation Discontinuity Location High Polynomial Degree 
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.



This work was supported by the Netherlands Organization for Scientific Research (NWO) and the European Union Marie Curie Cofund Action under Rubicon grant 680-50-1002.


  1. 1.
    Abgrall R (2010) A simple, Flexible and generic deterministic approach to uncertainty quantifications in nonlinear problems: application to fluid flow problems. In: Proceedings of the 5th European conference on computational fluid dynamics, ECCOMAS CFD, Lisbon, PortugalGoogle Scholar
  2. 2.
    Agarwal N, Aluru NR (2009) A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainty. Journal of Computational Physics 228: 7662–7688MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Babuška I, Tempone R, Zouraris GE (2004) Galerkin finite elements approximation of stochastic finite elements. SIAM Journal on Numerical Analysis 42: 800–825MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Babuška I, Nobile F, Tempone R (2007) A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Journal on Numerical Analysis 45: 1005–1034MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barth T (2012) On the propagation of statistical model parameter uncertainty in CFD calculations. Theoretical and Computational Fluid Dynamics 26: 435–457CrossRefGoogle Scholar
  6. 6.
    Barth T (2011) UQ methods for nonlinear conservation laws containing discontinuities. In: AVT-193 Lecture Series on Uncertainty Quantification, RTO-AVT-VKI Short Course on Uncertainty Quantification, Stanford, CaliforniaGoogle Scholar
  7. 7.
    Chassaing J-C, Lucor D (2010) Stochastic investigation of flows about airfoils at transonic speeds. AIAA Journal 48: 938–950CrossRefGoogle Scholar
  8. 8.
    Chorin AJ, Marsden JE (1979) A mathematical introduction to fluid mechanics. Springer-Verlag, New YorkCrossRefzbMATHGoogle Scholar
  9. 9.
    Cook PH, McDonald MA (1979) Firmin MCP, Aerofoil RAE 2822 – pressure distributions, and boundary layer and wake measurements. Experimental data base for computer program assessment, AGARD report AR 138Google Scholar
  10. 10.
    Dwight RP, Witteveen JAS, Bijl H (this issue) Adaptive uncertainty quantification for computational fluid dynamics. In: Uncertainty quantification, Lecture notes in computational science and engineering, SpringerGoogle Scholar
  11. 11.
    Harten A, Osher S (1987) Uniformly high-order accurate nonoscillatory schemes I. SIAM Journal on Numerical Analysis 24: 279–309MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Harten A (1989) ENO schemes with subcell resolution. Journal of Computational Physics 83: 148–184MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Foo J, Wan X, Karniadakis GE (2008) The multi-element probabilistic collocation method (ME-PCM): error analysis and applications. Journal of Computational Physics 227: 9572–9595MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ghanem RG, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer-Verlag, New YorkCrossRefzbMATHGoogle Scholar
  15. 15.
    Ghosh D, Ghanem R (2008) Stochastic convergence acceleration through basis enrichment of polynomial chaos expansions. International Journal on Numerical Methods in Engineering. 73: 162–184MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gottlieb D, Xiu D (2008) Galerkin method for wave equations with uncertain coefficients. Communications in Computational Physics 3: 505–518MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ma X, Zabaras N (2009) An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. Journal of Computational Physics 228: 3084–3113MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Le Maître OP, Najm HN, Ghanem RG, Knio OM (2004) Multi–resolution analysis of Wiener–type uncertainty propagation schemes. Journal of Computational Physics 197: 502–531MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mathelin L, Le Maître OP (2007) Dual-based a posteriori error estimate for stochastic finite element methods. Communications in Applied Mathematics and Computational Science 2: 83–115MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Onorato G, Loeven GJA, Ghorbaniasl G, Bijl H, Lacor C (2010) Comparison of intrusive and non-intrusive polynomial chaos methods for CFD applications in aeronautics. In: Proceedings of the 5th European conference on computational fluid dynamics, ECCOMAS CFD, Lisbon, PortugalGoogle Scholar
  21. 21.
    Pettit CL, Beran PS (2006) Convergence studies of Wiener expansions for computational nonlinear mechanics. In: Proceedings of the 8th AIAA non-deterministic approaches conference, Newport, Rhode Island, AIAA-2006-1993Google Scholar
  22. 22.
    Poëtte G, Després B, Lucor D (2009) Uncertainty quantification for systems of conservation laws, Journal of Computational Physics 228: 2443–2467MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Simon F, Guillen P, Sagaut P, Lucor D (2010) A gPC-based approach to uncertain transonic aerodynamics. Computer Methods in Applied Mechanics and Engineering 199: 1091–1099MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Shu C-W, Osher S (1988) Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics 77: 439–471MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Shu C-W, Osher S (1989) Efficient implementation of essentially non-oscillatory shock-capturing schemes II, Journal of Computational Physics 83: 32–78MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sod GA (1978) A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. Journal of Computational Physics 27: 1–31MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Tryoen J, Le Maître O, Ndjinga M, Ern A (2010) Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems. Journal of Computational Physics 229: 6485–6511MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wan X, Karniadakis GE (2005) An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. Journal of Computational Physics 209: 617–642MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Witteveen JAS, Bijl H (2009) A TVD uncertainty quantification method with bounded error applied to transonic airfoil flutter. Communications in Computational Physics 6: 406–432MathSciNetCrossRefGoogle Scholar
  30. 30.
    Witteveen JAS, Loeven GJA, Bijl H (2009) An adaptive stochastic finite elements approach based on Newton-Cotes quadrature in simplex elements. Computers and Fluids 38: 1270–1288MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Witteveen JAS (2010) Second order front tracking for the Euler equations. Journal of Computational Physics 229: 2719–2739MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Witteveen JAS, Iaccarino G (2012) Simplex stochastic collocation with random sampling and extrapolation for nonhypercube probability spaces. SIAM Journal on Scientific Computing 34: A814–A838MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Witteveen JAS, Iaccarino G (2012) Refinement criteria for simplex stochastic collocation with local extremum diminishing robustness. SIAM Journal on Scientific Computing 34: A1522–A1543MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Witteveen JAS, Iaccarino G (2013) Simplex stochastic collocation with ENO-type stencil selection for robust uncertainty quantification. Journal of Computational Physics 239: 1–21CrossRefMathSciNetGoogle Scholar
  35. 35.
    Witteveen JAS, Iaccarino G (submitted) Subcell resolution in simplex stochastic collocation for spatial discontinuitiesGoogle Scholar
  36. 36.
    Xiu D, Karniadakis GE (2002) The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM Journal on Scientific Computing 24: 619–644MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Xiu D, Hesthaven JS (2005) High-order collocation methods for differential equations with random inputs. SIAM Journal on Scientific Computing 27: 1118–1139MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Center for Turbulence ResearchStanford UniversityStanfordUSA
  2. 2.Center for Mathematics and Computer Science (CWI)AmsterdamThe Netherlands
  3. 3.Mechanical EngineeringStanford UniversityStanfordUSA

Personalised recommendations