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Mathematical descriptions of flow phenomena usually come in the form ofpartial differential equations. The differential operators used in theseequations may have properties such as symmetry, skew-symmetry, positive ornegative (definite)-ness. Symmetry-preserving methods are such that thediscretized form of the continuous differential operator exhibits the sameproperties as the continuous operator itself. The use of symmetry-preservingdiscretizations makes it possible to construct discrete models which allow allthe manipulations needed to prove stability and (discrete) conservationproperties in the same way they were proven in the original continuous model.Furthermore, these methods allow a discretization of the continuous adjointwhich 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 arbitraryorder on curvilinear structured grids. The key idea is to use mutually-adjointsampling and interpolation operators to switch between the continuous anddiscrete operator. The novelty of this work is that it combines three importantrequirements for discretizations: first, the symmetry-preserving discretizationis made for arbitrary order of accuracy; second, the method works for everystructured curvilinear mesh; and third, the method can be applied to everycontinuous operator. This paper is the first in a series of papers thatgradually extends the theory to a general approach.

Symmetry-preserving discretizations of arbitrary order on structured curvilinear grids