Uncertainty Quantitification of conservation laws (application to compressible stochastic flows and open problems

Bruno Despres
Commissariat à l'Énergie Atomique (CEA)

Uncertainty quantification through stochastic spectral methods has been recently applied to several kinds of non-linear stochastic PDEs. We make a parallel with 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. 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.

We will discuss the application to compressible gas dynamics and review some open problems.

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