  Non-conforming numerical approximations offer increased flexibility for
applications that require high resolution in a localized area of the
computational domain or near complex geometries. Two key properties for
non-conforming methods to be applicable to real world applications are
conservation and energy stability. The summation-by-parts (SBP) property, which
certain finite-difference and discontinuous Galerkin methods have, finds
success for the numerical approximation of hyperbolic conservation laws,
because the proofs of energy stability and conservation can discretely mimic
the continuous analysis of partial differential equations. In addition, SBP
methods can be developed with high-order accuracy, which is useful for
simulations that contain multiple spatial and temporal scales. However,
existing non-conforming SBP schemes result in a reduction of the overall degree
of the scheme, which leads to a reduction in the order of the solution error.
This loss of degree is due to the particular interface coupling through a
simultaneous-approximation-term (SAT). We present in this work a novel class of
SBP-SAT operators that maintain conservation, energy stability, and have no
loss of the degree of the scheme for non-conforming approximations. The new
\emph{degree preserving} discretizations require an ansatz that the norm matrix
of the SBP operator is of a degree $\geq 2p$, in contrast to, for example,
existing finite difference SBP operators, where the norm matrix is $2p-1$
accurate. We demonstrate the fundamental properties of the new scheme with
rigorous mathematical analysis as well as numerical verification.
