Uncertainty quantification for systems of conservation laws

  • Authors:

  • Gaël Poëtte;Bruno Després;Didier Lucor

  • Affiliations:

  • Commissariat í l'ínergie Atomique, Centre DAM, Ile de France, Bruyères le Chítel, 91297 Arpajon Cedex, France and Institut Jean Le Rond D'Alembert Université Pierre et Mar ...;Commissariat í l'ínergie Atomique, Centre DAM, Ile de France, Bruyères le Chítel, 91297 Arpajon Cedex, France;Institut Jean Le Rond D'Alembert Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris, France

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2009

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Abstract

Uncertainty quantification through stochastic spectral methods has been recently applied to several kinds of non-linear stochastic PDEs. In this paper, we introduce a formalism based on kinetic theory to tackle uncertain hyperbolic systems of conservation laws with Polynomial Chaos (PC) methods. The idea is to introduce a new variable, the entropic variable, in bijection with our vector of unknowns, which we develop on the polynomial basis: by performing a Galerkin projection, we obtain a deterministic system of conservation laws. We state several properties of this deterministic system in the case of a general uncertain system of conservation laws. We then apply the method to the case of the inviscid Burgers' equation with random initial conditions and we present some preliminary results for the Euler system. We systematically compare results from our new approach to results from the stochastic Galerkin method. In the vicinity of discontinuities, the new method bounds the oscillations due to Gibbs phenomenon to a certain range through the entropy of the system without the use of any adaptative random space discretizations. It is found to be more precise than the stochastic Galerkin method for smooth cases but above all for discontinuous cases.