  We study non-conforming grid interfaces for summation-by-parts finite
difference methods applied to partial differential equations with second
derivatives in space. To maintain energy stability, previous efforts have been
forced to accept a reduction of the global convergence rate by one order, due
to large truncation errors at the non-conforming interface. We avoid the order
reduction by generalizing the interface treatment and introducing order
preserving interpolation operators. We prove that, given two diagonal-norm
summation-by-parts schemes, order preserving interpolation operators with the
necessary properties are guaranteed to exist, regardless of the grid-point
distributions along the interface. The new methods retain the stability and
global accuracy properties of the underlying schemes for conforming interfaces.
