  We study the numerical solutions of time-dependent systems of partial
differential equations, focusing on the implementation of boundary conditions.
The numerical method considered is a finite difference scheme constructed by
high order summation by parts operators, combined with a boundary procedure
using penalties (SBP-SAT).
  Recently it was shown that SBP-SAT finite difference methods can yield
superconvergent functional output if the boundary conditions are imposed such
that the discretization is dual consistent. We generalize these results so that
they include a broader range of boundary conditions and penalty parameters. The
results are also generalized to hold for narrow-stencil second derivative
operators. The derivations are supported by numerical experiments.
