A Roe variable based chaos method for the Euler equations under uncertainty Per Pettersson ∗,† , Gianluca Iaccarino ∗ and Jan Nordstr¨ om †,‡ ∗ Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA 94305, USA. † Department of Information Technology, Uppsala University, P.O. Box 337, SE-75105 Uppsala, Sweden. ‡ Department of Mathematics, Link¨ oping University, SE-58183 Link¨ oping, Sweden. Abstract The Euler equations subject to uncertainty in the input parameters are investigated via the stochastic Galerkin approach. We present a new fully intrusive method based on a variable transformation of the continuous equations. Roe variables are employed to get quadratic dependence in the flux function and a well-defined Roe average matrix that can be determined without matrix inversion. In previous formulations based on generalized chaos expansion of the physical vari- ables, the need to introduce stochastic expansions of inverse quantities, or square-roots of stochastic quantities of interest, adds to the number of possible different ways to approximate the original stochastic problem. We present a method where no auxiliary quantities are needed, resulting in an unambiguous problem formulation. The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, the Roe formulation is more robust and can handle cases of supersonic flow, for which the conservative variable formula- tion leads to instability. For more extreme cases, where the global Legendre polyno- mials poorly approximate discontinuities in stochastic space, we use the localized Haar wavelet basis. 1 Introduction Generalized chaos expansions are frequently used to represent uncertain quantities in numerical solutions of PDEs with uncertainty in e.g. initial- and boundary data, 1