Linear and quadratic invariants preserving discretization of Euler equations

In the context of the numerical treatment of convective terms in compressible
transport equations, general criteria for linear and quadratic invariants preservation, valid
on uniform and non-uniform (Cartesian) meshes, have been recently derived by using a
matrix-vector approach, for both finite-difference and finite-volume methods ([1, 2]). In
this work, which constitutes a follow-up investigation of the analysis presented in [1, 2],
this theory is applied to the spatial discretization of convective terms for the system of
Euler equations. A classical formulation already presented in the literature is investigated
and reformulated within the matrix-vector approach. The relations among the discrete
versions of the various terms in the Euler equations are analyzed and the additional degrees
of freedom identified by the proposed theory are investigated. Numerical simulations on
a classical test case are used to validate the theory and to assess the effectiveness of the
various formulations.