High order approximation of probabilistic shock profiles in hyperbolic conservation laws with uncertain initial data
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 3, pp. 807-835.

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.

DOI : 10.1051/m2an/2012060
Classification : 65C30, 65M08, 65M12, 65M70, 65M75
Mots clés : uncertainty quantification, hyperbolic conservation laws, probabilistic shock profile, stochastic Galerkin finite volume schemes
@article{M2AN_2013__47_3_807_0,
     author = {Schwab, Christoph and Tokareva, Svetlana},
     title = {High order approximation of probabilistic shock profiles in hyperbolic conservation laws with uncertain initial data},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {807--835},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {3},
     year = {2013},
     doi = {10.1051/m2an/2012060},
     mrnumber = {3056410},
     zbl = {1266.65008},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2012060/}
}
TY  - JOUR
AU  - Schwab, Christoph
AU  - Tokareva, Svetlana
TI  - High order approximation of probabilistic shock profiles in hyperbolic conservation laws with uncertain initial data
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2013
SP  - 807
EP  - 835
VL  - 47
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2012060/
DO  - 10.1051/m2an/2012060
LA  - en
ID  - M2AN_2013__47_3_807_0
ER  - 
%0 Journal Article
%A Schwab, Christoph
%A Tokareva, Svetlana
%T High order approximation of probabilistic shock profiles in hyperbolic conservation laws with uncertain initial data
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2013
%P 807-835
%V 47
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2012060/
%R 10.1051/m2an/2012060
%G en
%F M2AN_2013__47_3_807_0
Schwab, Christoph; Tokareva, Svetlana. High order approximation of probabilistic shock profiles in hyperbolic conservation laws with uncertain initial data. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 3, pp. 807-835. doi : 10.1051/m2an/2012060. http://www.numdam.org/articles/10.1051/m2an/2012060/

[1] E. Godlewski and P. Raviart, Hyperbolic systems of conservation laws. Ellipses Publ., Paris (1995). | Zbl

[2] R. Leveque, Numerical methods for conservation laws. Birkhäuser Verlag (1992). | MR | Zbl

[3] S. Mishra and Ch. Schwab, Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random intitial data. Math. Comput. 81 (2012) 1979-2018. | MR | Zbl

[4] S. Mishra, Ch. Schwab and J. Šukys, Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions. J. Comput. Phys. 231 (2012) 3365-3388. | MR

[5] S. Mishra, Ch. Schwab and S. Tokareva, Stochastic Finite Volume methods for uncertainty quantification in hyperbolic conservation laws. In preparation (2012).

[6] J. Troyen, O. Le Maître, M. Ndjinga and A. Ern, Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems. J. Comput. Phys. 229 (2010) 6485-6511. | MR | Zbl

[7] J. Troyen, O. Le Maître, M. Ndjinga and A. Ern, Roe solver with entropy corrector for uncertain hyperbolic systems. J. Comput. Phys. 235 (2010) 491-506. | MR | Zbl

[8] E. H. Lieb and M. Loss, Analysis: 2nd Ed. Amer. Math. Soc. Graduate Studies in Math. 14 (2001). | MR | Zbl

[9] O.G. Ernst, A. Mugler, H.J. Starkloff and E. Ullmann, On the convergence of generalized polynomial chaos expansions. ESAIM: M2AN 46 (2012) 317-339. | Numdam | MR | Zbl

[10] D. Xiu and G.E. Karniadakis, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4927-4948. | MR | Zbl

[11] D. Xiu and G.E. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187 (2003) 137-167. | MR | Zbl

[12] R. Abgrall, A simple, flexible and generic deterministic approach to uncertainty quantification in non-linear problems. Rapport de Recherche, INRIA 00325315 (2007).

[13] G. Poëtte, B. Després and D. Lucor, Uncertainty quantification for systems of conservation laws. J. Comput. Phys. 228 (2009) 2443-2467. | MR | Zbl

[14] R. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach. Dover (2003). | Zbl

[15] D. Gottlieb and D. Xiu, Galerkin method for wave equations with uncertain coefficients. Commun. Comput. Phys. 3 (2008) 505-518. | MR | Zbl

[16] G. Lin, C.-H. Su and G.E. Karniadakis, Predicting shock dynamics in the presence of uncertainties. J. Comput. Phys. 217 (2006) 260-276. | MR | Zbl

[17] G. Lin, C.-H. Su and G.E. Karniadakis, Stochastic modelling of random roughness in shock scattering problems: theory and simulations. Comput. Methods Appl. Mech. Eng. 197 (2008) 3420-3434. | MR | Zbl

[18] X. Wan and G.E. Karniadakis, Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28 (2006) 901-928. | MR | Zbl

[19] B. Debusschere, H. Najm, P. Pébay, O. Knio, R. Ghanem and O. Le Maître, Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J. Sci. Comput. 26 (2004) 698-719. | MR | Zbl

[20] O. Knio and O. Le Maître, Uncertainty propagation in CFD using polynomial chaos decomposition. Fluid. Dynam. Res. 38 (2006) 616-640. | MR | Zbl

[21] O. Le Maître, O. Knio, H. Najm and R. Ghanem, Uncertainty propagation using Wiener-Haar expansions. J. Comput. Phys. 197 (2004) 28-57. | MR | Zbl

[22] O. Le Maître, H. Najm, R. Ghanem and O. Knio, Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comput. Phys. 197 (2004) 502-531. | MR | Zbl

[23] O. Le Maître, H. Najm, P. Pébay, R. Ghanem and O. Knio, Multi-resolution analysis scheme for uncertainty quantification in chemical systems. SIAM J. Sci. Comput. 29 (2007) 864-889. | MR | Zbl

[24] T. Barth, On the propagation of the statistical model parameter uncertainty in CFD calculations. Theoret. Comput. Fluid Dyn. 26 (2012) 435-457. | Zbl

[25] C.W. Shu, High order ENO and WENO schemes for computational fluid dynamics. In High-Order Methods for Computational Phys. Springer 9 (1999). | MR | Zbl

Cité par Sources :