@TechReport{ it:2012-021,
author = {Per Pettersson and Gianluca Iaccarino and Jan
Nordstr{\"o}m},
title = {A {R}oe Variable Based Chaos Method for the {E}uler
Equations under Uncertainty},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2012},
number = {2012-021},
month = aug,
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 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 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 leads to instability. For
more extreme cases, where the global Legendre polynomials
poorly approximate discontinuities in stochastic space, we
use the localized Haar wavelet basis.},
note = {A compete rewrite with new results appears as report nr
2012-033.}
}