. Ment-riche, Il serait donc intéressant de construire un solveur d'ordre supérieur en espace, en utilisant par exemple des techniques de type WENO ou DG (assorties de limiteurs adéquats) Une alternative serait de contrôler la discrétisation spatiale en utilisant également une méthode adaptative

. Ensuite, des techniques d'enrichissement plus avancées ont besoin d'être développées pour traiter le cas de plus grandes dimensions stochastiques. Elles font l'objet de travaux en cours, Une analyse des temps de calcul est également à prévoir dans le cas de grandes dimensions stochastiques

. Enfin, il serait souhaitable de comparer notre méthode avec les autres méthodes existantes pour propager les incertitudes dans les systèmes hyperboliques stochastiques , par exemple avec celles de Abgrall, Poette

R. Abgrall and ?. , flexible and generic deterministic approach to uncertainty quantifications in non linear problems : application to fluid flow problems, J. Comput. Phys, 2008.
URL : https://hal.archives-ouvertes.fr/inria-00325315

M. Abramowitz and &. , Stegun ? Handbook of Mathematical Functions, 1970.

R. Agarwal and &. , Wong ? Error inequalities in polynomial interpolation and their applications, 1993.

R. Askey and &. , Wilson ? « Some basic hypergeometric polynomials that generalize Jacobi polynomials, pp.1-55, 1985.
DOI : 10.1090/memo/0319

I. Babu?kababu?ka, F. Nobile, and &. , A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data, SIAM Journal on Numerical Analysis, vol.45, issue.3, pp.1005-1034, 2007.
DOI : 10.1137/050645142

A. Barth, C. Schwab, and &. , Zollinger ? « Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients, Seminar for Applied Mathematics, ETH Zürich, 2010.

M. Berveiller, B. Sudret, and &. , Lemaire ? « Stochastic finite element : a non intrusive approach by regression », Eur, J. Comput. Mech, vol.15, pp.81-92, 2006.

R. Cameron and &. , The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals, The Annals of Mathematics, vol.48, issue.2, pp.385-392, 1947.
DOI : 10.2307/1969178

C. Clenshaw and &. , Curtis ? « A method for numerical integration on an automatic computer, pp.197-205, 1960.

A. Cohen, Wavelet methods in numerical analysis, Studies in Mathematics and its Applications, 2003.
DOI : 10.1016/S1570-8659(00)07004-6

A. Cohen, S. Kaber, S. Müller, and &. , Fully adaptive multiresolution finite volume schemes for conservation laws, Mathematics of Computation, vol.72, issue.241, pp.183-225, 2001.
DOI : 10.1090/S0025-5718-01-01391-6

S. Das, R. Ghanem, and &. , Asymptotic Sampling Distribution for Polynomial Chaos Representation from Data: A Maximum Entropy and Fisher Information Approach, SIAM Journal on Scientific Computing, vol.30, issue.5, pp.2207-2234, 2008.
DOI : 10.1137/060652105

M. Deb, I. Babuska, and &. , Solution of stochastic partial differential equations using Galerkin finite element techniques, Computer Methods in Applied Mechanics and Engineering, vol.190, issue.48, pp.6359-6372, 2001.
DOI : 10.1016/S0045-7825(01)00237-7

B. Debusschere, H. Najm, A. Matta, O. Knio, and R. Ghanem, Protein labeling reactions in electrochemical microchannel flow: Numerical simulation and uncertainty propagation, Physics of Fluids, vol.15, issue.8, pp.2238-2250, 2003.
DOI : 10.1063/1.1582857

B. Debusschere, H. Najm, P. Pébay, O. Knio, and R. Ghanem, Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes, SIAM Journal on Scientific Computing, vol.26, issue.2, pp.698-719, 2004.
DOI : 10.1137/S1064827503427741

P. Degond, P. Peyrard, G. Russo, and &. , Polynomial upwind schemes for hyperbolic systems, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.328, issue.6, pp.479-483, 1999.
DOI : 10.1016/S0764-4442(99)80194-3
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.144.6745

G. Deodatis and &. , Shinozuka ? « Weighted integral method, J. Engrg. Math, vol.117, issue.8, pp.1865-1877, 1991.

C. Desceliers, R. Ghanem, and &. , Maximum likelihood estimation of stochastic chaos representations from experimental data, International Journal for Numerical Methods in Engineering, vol.11, issue.6, pp.978-1001, 2006.
DOI : 10.1002/nme.1576
URL : https://hal.archives-ouvertes.fr/hal-00686154

C. Desceliers, C. Soize, &. R. Ghanem, and ?. , Identification of Chaos Representations of Elastic Properties of Random Media Using Experimental Vibration Tests, Computational Mechanics, vol.60, issue.5, pp.831-838, 2007.
DOI : 10.1007/s00466-006-0072-7
URL : https://hal.archives-ouvertes.fr/hal-00686150

D. A. Di-pietro and &. A. , Ern ? Mathematical Aspects of Discontinuous Galerkin Methods, 2011.

O. Ditlevsen and &. , Tarp-Johansen ? « Choice of input fields in stochastic finite-elements, Probabilistic Engineering Mechanics, pp.63-72, 1999.

F. Dubois and &. , A non-parameterized entropy correction for Roe's approximate Riemann solver, Numerische Mathematik, vol.73, issue.2, pp.169-208, 1996.
DOI : 10.1007/s002110050190

A. Ern and &. , Guermond ? Theory and Practice of Finite Elements, 2004.

B. Ganapathysubramanian and &. , Sparse grid collocation schemes for stochastic natural convection problems, Journal of Computational Physics, vol.225, issue.1, pp.652-685, 2007.
DOI : 10.1016/j.jcp.2006.12.014

L. Ge, K. Cheung, and &. , Kobayashi ? « Stochastic solution for uncertainty propagation in nonlinear shallow-water Equations, Journal of Hydraulic Engineering, pp.1732-1743, 2008.

T. Gerstner and &. , Griebel ? « Numerical integration using sparse grids », Numer, Numerical Algorithms, vol.18, issue.3/4, pp.209-232, 1998.
DOI : 10.1023/A:1019129717644

R. Ghanem and &. , Spanos ? Stochastic Finite Elements : A Spectral Approach, pp.7-20, 2003.

E. Godlewski and &. , Raviart ? Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, vol.118, pp.22-43, 1996.

D. Gottlieb and &. , Xiu ? « Galerkin method for wave equations with uncertain coefficients, Commun. Comput. Phys, vol.3, issue.2, pp.505-518, 2008.

J. S. Hesthaven and &. , Warburton ? Nodal discontinuous Galerkin methods : algorithms, analysis, and applications, 2008.
DOI : 10.1007/978-0-387-72067-8

N. Hovhannisyan, S. Müller, and &. , Schäfer ? « Adaptive multiresolution discontinuous galerkin schemes for conservation laws, 2010.
DOI : 10.1090/s0025-5718-2013-02732-9

A. Keese and &. , Matthies ? « Numerical methods and Smolyak quadrature for nonlinear stochastic partial differential equations, 2003.

M. Kleiber and &. , Hien ? The Stochastic Finite Element Method, 1992.

O. Knio, Uncertainty propagation in CFD using polynomial chaos decomposition, Fluid Dynamics Research, vol.38, issue.9, pp.616-640, 2006.
DOI : 10.1016/j.fluiddyn.2005.12.003

O. and L. Maître, Knio ? Spectral Methods for Uncertainty Quantification, Scientific Computation With applications to computational fluid dynamics, 2010.

O. Le-maître, O. Knio, H. Najm, and &. , A Stochastic Projection Method for Fluid Flow, Journal of Computational Physics, vol.173, issue.2, pp.481-511, 2001.
DOI : 10.1006/jcph.2001.6889

O. Le-maître, O. Knio, H. Najm, and &. , Uncertainty propagation using Wiener???Haar expansions, Journal of Computational Physics, vol.197, issue.1, pp.28-57, 2004.
DOI : 10.1016/j.jcp.2003.11.033

O. Le-maître, H. Najm, and R. Ghanem, Multi-resolution analysis of Wiener-type uncertainty propagation schemes, Journal of Computational Physics, vol.197, issue.2, pp.502-531, 2004.
DOI : 10.1016/j.jcp.2003.12.020

O. Le-maître, H. Najm, P. Pébay, and R. Ghanem, Multi???Resolution???Analysis Scheme for Uncertainty Quantification in Chemical Systems, SIAM Journal on Scientific Computing, vol.29, issue.2, pp.864-889, 2007.
DOI : 10.1137/050643118

O. , L. Maître, M. Reagan, H. Najm, and R. Ghanem, Knio ? « A stochastic projection method for fluid flow. II. Random process, J. Comput. Phys, vol.181, pp.9-44, 2002.

G. Lin, C. Su, and &. , Predicting shock dynamics in the presence of uncertainties, Journal of Computational Physics, vol.217, issue.1, pp.260-276, 2006.
DOI : 10.1016/j.jcp.2006.02.009

G. Lin, C. Su, and &. , Karniadakis ? « Stochastic modeling of random roughness in shock scattering problems : theory and simulations, Comput. Methods Appl. Mech. Engrg, vol.197, pp.43-44, 2008.

X. Liu, S. Osher, and &. , Weighted Essentially Non-oscillatory Schemes, Journal of Computational Physics, vol.115, issue.1, pp.200-212, 1994.
DOI : 10.1006/jcph.1994.1187
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.8744

X. Ma and &. , An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations, Journal of Computational Physics, vol.228, issue.8, pp.3084-3113, 2009.
DOI : 10.1016/j.jcp.2009.01.006

L. Mathelin and &. , Hussaini ? « A stochastic collocation algorithm for uncertainty analysis, 2003.

L. Mathelin, M. Hussaini, and &. , Stochastic approaches to uncertainty quantification in CFD simulations, Numerical Algorithms, vol.187, issue.2, pp.209-236, 2005.
DOI : 10.1007/BF02810624

M. Mckay, W. Conover, and &. , Beckman ? « A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, vol.21, pp.239-245, 1979.

S. Mishra and &. , Schwab ? « Sparse tensor multi-level Monte Carlo Finite Volume methods for hyperbolic conservation laws with random initial data, Seminar for Applied Mathematics, ETH Zürich, pp.6-26, 2010.

W. Morokoff, &. R. Caflisch, and ?. Carlo-integration, Quasi-Monte Carlo Integration, Journal of Computational Physics, vol.122, issue.2, pp.218-230, 1995.
DOI : 10.1006/jcph.1995.1209
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.56.1652

S. Müller and ?. , Adaptive multiscale schemes for conservation laws, Lecture Notes in Computational Science and Engineering, vol.27, 2003.
DOI : 10.1007/978-3-642-18164-1

H. Najm and ?. , Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics, Annual Review of Fluid Mechanics, vol.41, issue.1, pp.35-52, 2009.
DOI : 10.1146/annurev.fluid.010908.165248

H. Najm, B. Debusschere, Y. Marzouk, and S. Widmer, Uncertainty quantification in chemical systems, International Journal for Numerical Methods in Engineering, vol.10, issue.2, pp.789-814, 2009.
DOI : 10.1002/nme.2551

M. Ndjinga and ?. , Computing the matrix sign and absolute value functions, Comptes Rendus Mathematique, vol.346, issue.1-2, pp.119-124, 2008.
DOI : 10.1016/j.crma.2007.11.028

F. Nobile, R. Tempone, and &. , An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data, SIAM Journal on Numerical Analysis, vol.46, issue.5, pp.2411-2442, 2008.
DOI : 10.1137/070680540

G. Poette, B. Després, and &. , Uncertainty quantification for systems of conservation laws, Journal of Computational Physics, vol.228, issue.7, pp.2443-2467, 2009.
DOI : 10.1016/j.jcp.2008.12.018

M. Reagan, H. Najm, and R. Ghanem, Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection, Combustion and Flame, vol.132, issue.3, pp.545-555, 2003.
DOI : 10.1016/S0010-2180(02)00503-5

M. Shinozuka and &. , Deodatis ? « Response variability of stochastic finite element systems, J. Engrg. Math, vol.114, issue.3, pp.499-519, 1988.

S. Smolyak and ?. , Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR, vol.4, pp.240-243, 1963.

C. Soize and ?. , A comprehensive overview of a non-parametric probabilistic approach of model uncertainties for predictive models in structural dynamics, Journal of Sound and Vibration, vol.288, issue.3, pp.623-652, 2005.
DOI : 10.1016/j.jsv.2005.07.009
URL : https://hal.archives-ouvertes.fr/hal-00686182

C. Soize, &. R. Ghanem, and ?. , Reduced Chaos decomposition with random coefficients of vector-valued random variables and random fields, Computer Methods in Applied Mechanics and Engineering, vol.198, issue.21-26
DOI : 10.1016/j.cma.2008.12.035
URL : https://hal.archives-ouvertes.fr/hal-00684487

J. Stoer and &. , Bulirsch ? Introduction to Numerical Analysis, 3ème éd, Texts in Applied Mathematics, pp.12-18, 2002.

E. Toro and ?. , Riemann solvers and numerical methods for fluid dynamics : A practical introduction, 2ème éd, pp.22-43, 1999.

J. Tryoen, O. Le-maître, and &. A. , Ern ? « Adaptive Anisotropic Stochastic Discretization Schemes for Uncertain Conservation Laws, Proceedings of the ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting, 2010.

J. Tryoen, O. Le-maître, M. Ndjinga, and &. , Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems, Journal of Computational Physics, vol.229, issue.18, pp.6485-6511, 2010.
DOI : 10.1016/j.jcp.2010.05.007

X. Wan and &. , Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures, SIAM Journal on Scientific Computing, vol.28, issue.3, pp.901-928, 2006.
DOI : 10.1137/050627630

N. Wiener, ?. The-homogeneous-chaos, and ». Amer, The Homogeneous Chaos, American Journal of Mathematics, vol.60, issue.4, pp.897-936, 1938.
DOI : 10.2307/2371268

D. Xiu and &. , High-Order Collocation Methods for Differential Equations with Random Inputs, SIAM Journal on Scientific Computing, vol.27, issue.3, pp.1118-1139, 2005.
DOI : 10.1137/040615201

D. Xiu and &. , The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations, SIAM Journal on Scientific Computing, vol.24, issue.2, pp.619-644, 2002.
DOI : 10.1137/S1064827501387826

F. Yamazaki, M. Shinozuka, and &. , Dasgupta ? « Neumann expansion for stochastic finite element analysis, J. Engrg. Math, vol.114, issue.8, pp.1335-1354, 1988.