@TechReport{	  it:2012-025,
  author	= {Jens Berg and Jan Nordstr{\"o}m},
  title		= {On the Impact of Boundary Conditions on Dual Consistent
		  Finite Difference Discretizations},
  institution	= {Department of Information Technology, Uppsala University},
  department	= {Division of Scientific Computing},
  year		= {2012},
  number	= {2012-025},
  month		= sep,
  abstract	= {In this paper we derive well-posed boundary conditions for
		  a linear incompletely parabolic system of equations, which
		  can be viewed as a model problem for the compressible
		  Navier-Stokes equations. We show a general procedure for
		  the construction of the boundary conditions such that both
		  the primal and dual equations are well-posed. The form of
		  the boundary conditions is chosen such that reduction to
		  first order form with its complications can be avoided.
		  
		  The primal equation is discretized using finite difference
		  operators on summation-by-parts form with weak boundary
		  conditions. It is shown that the discretization can be made
		  energy stable, and that energy stability is sufficient for
		  dual consistency. Since reduction to first order form can
		  be avoided, the discretization is significantly simpler
		  compared to a discretization using Dirichlet boundary
		  conditions.
		  
		  We compare the new boundary conditions with standard
		  Dirichlet boundary conditions in terms of rate of
		  convergence, errors and discrete spectra. It is shown that
		  the scheme with the new boundary conditions is not only far
		  simpler, but have smaller errors, error bounded properties,
		  and highly optimizable eigenvalues, while maintaining all
		  desirable properties of a dual consistent discretization.}
}