TY - JOUR T1 - Entropies and Symmetrization of Hyperbolic Stochastic Galerkin Formulations AU - Gerster , Stephan AU - Herty , Michael JO - Communications in Computational Physics VL - 3 SP - 639 EP - 671 PY - 2020 DA - 2020/02 SN - 27 DO - http://doi.org/10.4208/cicp.OA-2019-0047 UR - https://global-sci.org/intro/article_detail/cicp/13925.html KW - Hyperbolic partial differential equations, uncertainty quantification, stochastic Galerkin, shallow water equations, well-posedness, entropy, Roe variable transform. AB -

Stochastic quantities of interest are expanded in generalized polynomial chaos expansions using stochastic Galerkin methods. An application of hyperbolic differential equations in general does 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, well-posedness and energy estimates.