Multi-resolution and adaptive stochastic Galerkin methods for uncertain conservation equations.

We present a multi-resolution scheme, based on piecewise polynomial approximations at the stochastic level, for the propagation of parametric uncertainties in hyperbolic systems of conservation laws. The numerical method rely on a Galerkin projection technique at the stochastic level, with a finite-volume discretization and a Roe-type solver (with entropy corrector) in space and time. emergence of shocks and discontinuities in the solution with respect to the uncertain parameters. However, these features remain localized along singularity curves which may themselves have uncertain locations. We propose to use multi-resolution schemes coupled with adaptive strategies for the stochastic discretization of the solution. In particular, we will detail an anisotropic adaptive strategy where the stochastic discretization is adapted locally in space and time. Examples of applications and efficiency / complexity assessment of the method will be shown.