Entropies and symmetrization of hyperbolic stochastic Galerkin formulations

Abstract

Stochastic quantities of interest are expanded in generalized polynomial chaos expansions using stochastic Galerkin methods. An application to hyperbolic differential equations does in general not transfer hyperbolicity to the coefficients of the truncated series expansion. For the Haar basis and for piecewise linear multiwavelets we present convex entropies for the systems of coefficients of the one-dimensional shallow water equations by using the Roe variable transform. This allows to obtain hyperbolicity, wellposedness and energy estimates.