Adaptive Anisotropic Spectral Stochastic Methods for Uncertain Scalar Conservation Laws

Julie Tryoen (1, 2) , Olivier Le Maitre (2) , 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

This paper deals with the design of adaptive anisotropic discretization schemes for conservation laws with stochastic parameters. A Finite Volume scheme is used for the deterministic discretization, while a piecewise polynomial representation is used at the stochastic level. The methodology is designed in the context of intrusive Galerkin projection methods with Roe-type solver. The adaptation aims at selecting the stochastic resolution level based on the local smoothness of the solution in the stochastic domain. In addition, the stochastic features of the solution greatly vary in the space and time so that the constructed stochastic approximation space depends on space and time. The dynamically evolving stochastic discretization uses a tree-structure representation that allows for the efficient implementation of the various operators needed to perform anisotropic multiresolution analysis. Efficiency of the overall adaptive scheme is assessed on the stochastic traffic equation with uncertain initial conditions and velocity leading to expansion waves and shocks that propagate with random velocities. Numerical tests highlight the computational savings achieved as well as the benefit of using anisotropic discretizations in view of dealing with problems involving a larger number of stochastic parameters.

Mots clés: uncertainty quantification; stochastic multiresolution; stochastic spectral method; adaptivity; Galerkin projection; conservation laws; hyperbolic systems

Domaines: Analyse numérique [math.NA]

Julie Tryoen, Olivier Le Maitre, Alexandre Ern. Adaptive Anisotropic Spectral Stochastic Methods for Uncertain Scalar Conservation Laws. 2012. ⟨hal-00681200⟩