Uncertainty Propagation for Systems of Conservation Laws, High Order Stochastic Spectral Methods

The application of the stochastic Galerkin-generalized Polynomial Chaos approach (sG-gPC) (Wiener, Am. J. Math. 60:897–936, 1938; Cameron and Martin, Ann. Math. 48:385–392, 1947; Xiu and Karniadakis, SIAM J. Sci. Comp. 24(2):619–644, 2002) for Uncertainty Propagation through NonLinear Systems of Conservation Laws (SLC) is known to encounter several difficulties: dimensionality (see, e.g., Nobile et al., SIAM J. Numer. Anal. 46(5):2309–2345, 2008; Blatman and Sudret, C. R. Mec. 336:518–523, 2008; Witteveen and Bijl, Comp. Struct. 86(23–24):2123–2140, 2008), non linearities (see, e.g., Debusshere et al., J. Sci. Comp. 26:698–719, 2004; Witteveen and Bijl, 47th AIAA Aerospace Sciences Meeting and Exhibit, 2006–2066, 2006), discontinuities (see Wan and Karniadakis, SIAM J. Sci. Comp. 27(1–3), 2006; Lin et al., J. Comp. Phys. 217:260–276, 2006; Le Maitre and Knio, J. Comp. Phys. 197:28–57, 2004; Le Maitre et al., J. Comp. Phys. 197:502–531, 2004; Abgrall, Rapport de Recherche INRIA, 2007). In this paper, we first illustrate on a simple SLC (p-system) the difficulties occuring when dealing with non linearities and discontinuities. We will then present a new non adaptive high order uncertainty propagation method based on the entropy of the system of conservation laws, efficient on NonLinear systems and discontinuous solutions. Convergence tests are performed and spectral convergence is reached.

[1]  Rémi Abgrall.  A simple, flexible and generic deterministic approach to uncertainty quantifications in non linear problems: application to fluid flow problems , 2008 .

[2]  R. Ghanem,et al.  Uncertainty propagation using Wiener-Haar expansions , 2004 .

[3]  Habib N. Najm,et al.  Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes , 2005, SIAM J. Sci. Comput..

[4]  L. Mead,et al.  Maximum entropy in the problem of moments , 1984 .

[5]  Bruno Després,et al.  Uncertainty quantification for systems of conservation laws , 2009, J. Comput. Phys..

[6]  Guang Lin,et al.  Predicting shock dynamics in the presence of uncertainties , 2006, J. Comput. Phys..

[7]  G. Karniadakis,et al.  Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures , 2006, SIAM J. Sci. Comput..

[8]  Michael Junk,et al.  MAXIMUM ENTROPY FOR REDUCED MOMENT PROBLEMS , 2000 .

[9]  George E. Karniadakis,et al.  Beyond Wiener–Askey Expansions: Handling Arbitrary PDFs , 2006, J. Sci. Comput..

[10]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[11]  R. Ghanem,et al.  Multi-resolution analysis of wiener-type uncertainty propagation schemes , 2004 .

[12]  Dongbin Xiu,et al.  Discontinuity detection in multivariate space for stochastic simulations , 2009, J. Comput. Phys..

[13]  W. T. Martin,et al.  The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals , 1947 .