@TechReport{	  it:2012-033,
  author	= {Per Pettersson and Gianluca Iaccarino and Jan
		  Nordstr{\"o}m},
  title		= {A Stochastic {G}alerkin Method for the {E}uler Equations
		  with {R}oe Variable Transformation},
  institution	= {Department of Information Technology, Uppsala University},
  department	= {Division of Scientific Computing},
  year		= {2012},
  number	= {2012-033},
  month		= nov,
  note		= {This is a complete rewrite of report nr 2012-021 with new
		  results. A more general framework for the representation of
		  uncertainty is used. All figures have been replaced and
		  more numerical results have been added (methods of
		  manufactured solutions, convergence in space and the
		  stochastic dimension for subsonic and supersonic flow).},
  abstract	= {The Euler equations subject to uncertainty in the initial
		  and boundary conditions 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 polynomial
		  chaos expansion of the physical variables, 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 the
		  square roots occur in the choice of variables and 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 formulation fails to produce a
		  bounded solution. We use a multi-wavelet basis that can be
		  chosen to include a large number of resolution levels to
		  handle more extreme cases (e.g. strong discontinuities) in
		  a robust way. For smooth cases, the order of the polynomial
		  representation can be increased for increased accuracy.}
}