Intrusive projection methods with upwinding for uncertain nonlinear hyperbolic systems

Julie Tryoen (1, 2) , Olivier Le Maitre (2, 3) , Michael Ndjinga (4) , Alexandre Ern (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 LIMSI - Laboratoire d'Informatique pour la Mécanique et les Sciences de l'Ingénieur
3 DM2S - Département de Modélisation des Systèmes et Structures
4 SGLS - Service de Génie Logiciel pour la Simulation

This paper deals with spectral stochastic methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws. We consider problems with parametric uncertainty in initial conditions and model coefficients, whose solutions exhibit discontinuities in physical as well as in stochastic spaces. The spectral stochastic method relies on multi-resolution schemes with multi-wavelet or local polynomial bases. A Galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution. Hyperbolicity of the resulting Galerkin system is analyzed. A finite volume scheme with a Roe-type solver is used for discretization in physical space and time. An original technique is introduced for the fast evaluation of approximate upwind matrices, which is particularly well adapted to local polynomial bases. Efficiency and robustness of the overall method are assessed on the Burgers and Euler equations with shocks.

Mots clés: Uncertainty Quantification; hyperbolic systems; conservation laws; spectral stochastic methods; Multi-Resolution; upwinding

Domaines: Analyse numérique [math.NA]

Julie Tryoen, Olivier Le Maitre, Michael Ndjinga, Alexandre Ern. Intrusive projection methods with upwinding for uncertain nonlinear hyperbolic systems. Journal of Computational Physics, 2010, 229 (18), pp.6485-6511. ⟨10.1016/j.jcp.2010.05.007⟩. ⟨hal-00375616v2⟩