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High order approximation of probabilistic shock profiles in hyperbolic conservation laws with uncertain initial data∗∗∗

Published online by Cambridge University Press:  29 March 2013

Christoph Schwab  and
Svetlana Tokareva
Christoph Schwab
Affiliation:
ETH Zürich, Seminar for Applied Mathematics, Zürich, Switzerland. christoph.schwab@sam.math.ethz.ch; svetlana.tokareva@sam.math.ethz.ch
Svetlana Tokareva
Affiliation:
ETH Zürich, Seminar for Applied Mathematics, Zürich, Switzerland. christoph.schwab@sam.math.ethz.ch; svetlana.tokareva@sam.math.ethz.ch
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Abstract

We analyze the regularity of random entropy solutions to scalar hyperbolic conservation laws with random initial data. We prove regularity theorems for statistics of random entropy solutions like expectation, variance, space-time correlation functions and polynomial moments such as gPC coefficients. We show how regularity of such moments (statistical and polynomial chaos) of random entropy solutions depends on the regularity of the distribution law of the random shock location of the initial data. Sufficient conditions on the law of the initial data for moments of the random entropy solution to be piece-wise smooth functions of space and time are identified, even in cases where path-wise random entropy solutions are discontinuous almost surely. We extrapolate the results to hyperbolic systems of conservation laws in one space dimension. We then exhibit a class of stochastic Galerkin discretizations which allows to derive closed deterministic systems of hyperbolic conservation laws for the coefficients in truncated polynomial chaos expansions of the random entropy solution. Based on the regularity theory developed here, we show that depending on the smoothness of the law of the initial data, arbitrarily high convergence rates are possible for the computation of coefficients in gPC approximations of random entropy solutions for Riemann problems with random shock location by combined Stochastic Galerkin Finite Volume schemes.

Keywords

Uncertainty quantificationhyperbolic conservation lawsprobabilistic shock profilestochastic Galerkin finite volume schemes
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 3 , May 2013 , pp. 807 - 835
Copyright
© EDP Sciences, SMAI, 2013

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References

E. Godlewski and P. Raviart, Hyperbolic systems of conservation laws. Ellipses Publ., Paris (1995).
R. LeVeque, Numerical methods for conservation laws. Birkhäuser Verlag (1992).
Mishra, S. and Schwab, Ch., Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random intitial data. Math. Comput. 81 (2012) 1979–2018. Google Scholar
Mishra, S., Schwab, Ch. and Šukys, J., Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions. J. Comput. Phys. 231 (2012) 3365–3388. Google Scholar
S. Mishra, Ch. Schwab and S. Tokareva, Stochastic Finite Volume methods for uncertainty quantification in hyperbolic conservation laws. In preparation (2012).
Troyen, J., Le Maître, O., Ndjinga, M. and Ern, A., Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems. J. Comput. Phys. 229 (2010) 6485–6511. Google Scholar
Troyen, J., Le Maître, O., Ndjinga, M. and Ern, A., Roe solver with entropy corrector for uncertain hyperbolic systems. J. Comput. Phys. 235 (2010) 491–506. Google Scholar
E. H. Lieb and M. Loss, Analysis: 2nd Ed. Amer. Math. Soc. Graduate Studies in Math. 14 (2001).
Ernst, O.G., Mugler, A., Starkloff, H.J. and Ullmann, E., On the convergence of generalized polynomial chaos expansions. ESAIM: M2AN 46 (2012) 317–339. Google Scholar
Xiu, D. and Karniadakis, G.E., Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4927–4948. Google Scholar
Xiu, D. and Karniadakis, G.E., Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187 (2003) 137–167. Google Scholar
R. Abgrall, A simple, flexible and generic deterministic approach to uncertainty quantification in non-linear problems. Rapport de Recherche, INRIA 00325315 (2007).
Poëtte, G., Després, B. and Lucor, D., Uncertainty quantification for systems of conservation laws. J. Comput. Phys. 228 (2009) 2443–2467. Google Scholar
R. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach. Dover (2003).
Gottlieb, D. and Xiu, D., Galerkin method for wave equations with uncertain coefficients. Commun. Comput. Phys. 3 (2008) 505–518. Google Scholar
Lin, G., Su, C.-H. and Karniadakis, G.E., Predicting shock dynamics in the presence of uncertainties. J. Comput. Phys. 217 (2006) 260–276. Google Scholar
Lin, G., Su, C.-H. and Karniadakis, G.E., Stochastic modelling of random roughness in shock scattering problems: theory and simulations. Comput. Methods Appl. Mech. Eng. 197 (2008) 3420–3434. Google Scholar
Wan, X. and Karniadakis, G.E., Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28 (2006) 901–928. Google Scholar
Debusschere, B., Najm, H., Pébay, P., Knio, O., Ghanem, R. and Le Maître, O., Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J. Sci. Comput. 26 (2004) 698–719. Google Scholar
Knio, O. and Le Maître, O., Uncertainty propagation in CFD using polynomial chaos decomposition. Fluid. Dynam. Res. 38 (2006) 616–640. Google Scholar
Le Maître, O., Knio, O., Najm, H. and Ghanem, R., Uncertainty propagation using Wiener-Haar expansions. J. Comput. Phys. 197 (2004) 28–57. Google Scholar
Le Maître, O., Najm, H., Ghanem, R. and Knio, O., Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comput. Phys. 197 (2004) 502–531. Google Scholar
Le Maître, O., Najm, H., Pébay, P., Ghanem, R. and Knio, O., Multi-resolution analysis scheme for uncertainty quantification in chemical systems. SIAM J. Sci. Comput. 29 (2007) 864–889. Google Scholar
Barth, T., On the propagation of the statistical model parameter uncertainty in CFD calculations. Theoret. Comput. Fluid Dyn. 26 (2012) 435–457. Google Scholar
C.W. Shu, High order ENO and WENO schemes for computational fluid dynamics. In High-Order Methods for Computational Phys. Springer 9 (1999).