  Mathematical descriptions of flow phenomena usually come in the form of
partial differential equations. The differential operators used in these
equations may have properties such as symmetry, skew-symmetry, positive or
negative (definite)-ness. Symmetry-preserving methods are such that the
discretized form of the continuous differential operator exhibits the same
properties as the continuous operator itself. The use of symmetry-preserving
discretizations makes it possible to construct discrete models which allow all
the manipulations needed to prove stability and (discrete) conservation
properties in the same way they were proven in the original continuous model.
Furthermore, these methods allow a discretization of the continuous adjoint
which is at the same time the discrete adjoint of the discrete forward model.
Such adjoint models are not harder to code than the discrete forward model.
  This paper presents a new symmetry-preserving discretization of arbitrary
order on curvilinear structured grids. The key idea is to use mutually-adjoint
sampling and interpolation operators to switch between the continuous and
discrete operator. The novelty of this work is that it combines three important
requirements for discretizations: first, the symmetry-preserving discretization
is made for arbitrary order of accuracy; second, the method works for every
structured curvilinear mesh; and third, the method can be applied to every
continuous operator. This paper is the first in a series of papers that
gradually extends the theory to a general approach.
