@TechReport{	  it:2012-028,
  author	= {Per Pettersson and Alireza Doostan and Jan Nordstr{\"o}m},
  title		= {On Stability and Monotonicity Requirements of Discretized
		  Stochastic Conservation Laws with Random Viscosity},
  institution	= {Department of Information Technology, Uppsala University},
  department	= {Division of Scientific Computing},
  year		= {2012},
  number	= {2012-028},
  month		= sep,
  abstract	= {The stochastic Galerkin and collocation methods are used
		  to solve an advection-diffusion equation with uncertain and
		  spatially varying viscosity. We investigate well-posedness,
		  monotonicity and stability for the extended system
		  resulting from the Galerkin projection of the
		  advection-diffusion equation onto the stochastic basis
		  functions. High-order summation-by-parts operators and weak
		  imposition of boundary conditions are used to prove
		  stability of the semi-discrete system.
		  
		  It is essential that the eigenvalues of the resulting
		  viscosity matrix of the stochastic Galerkin system are
		  positive and we investigate conditions for this to hold.
		  When the viscosity matrix is diagonalizable, stochastic
		  Galerkin and stochastic collocation are similar in terms of
		  computational cost, and for some cases the accuracy is
		  higher for stochastic Galerkin provided that monotonicity
		  requirements are met. We also investigate the total spatial
		  operator of the semi-discretized system and its impact on
		  the convergence to steady-state.}
}