@TechReport{	  it:2013-016,
  author	= {Ken Mattsson and Martin Almquist and Mark H. Carpenter},
  title		= {Optimal Diagonal-Norm {SBP} Operators},
  institution	= {Department of Information Technology, Uppsala University},
  department	= {Division of Scientific Computing},
  year		= {2013},
  number	= {2013-016},
  month		= sep,
  abstract	= {Optimal boundary closures are derived for first
		  derivative, finite difference operators of order $2$, $4$,
		  $6$ and $8$. The closures are based on a diagonal-norm
		  summation-by-parts (SBP) framework, thereby guaranteeing
		  linear stability on piecewise curvilinear multi-block grids
		  and entropy stability for nonlinear equations that support
		  a convex extension. The new closures are developed by
		  enriching conventional approaches with additional boundary
		  closure stencils and non-equidistant grid distributions at
		  the domain boundaries. Greatly improved accuracy is
		  achieved near the boundaries, as compared with traditional
		  diagonal norm operators of the same order. The superior
		  accuracy of the new optimal diagonal-norm SBP operators is
		  demonstrated for linear hyperbolic systems in one dimension
		  and for the nonlinear compressible Euler equations in two
		  dimensions.}
}