Comparative advantages of compressible Euler formulations

Determine whether any one formulation of the compressible Euler equations offers advantages over the others, including formulations distinguished by their thermodynamic prognostic variables—such as potential temperature, entropy, internal energy, or total energy—and by their momentum-equation form, such as conservative or vector-invariant form.

Background

The paper surveys several formulations of the compressible Euler equations used in modern atmospheric dynamical cores. These formulations differ both in the choice of thermodynamic prognostic variable and in the representation of the momentum equation. The paper develops an entropy-stable, energy-conservative, and well-balanced discontinuous Galerkin method for the vector-invariant formulation with potential temperature as the prognostic thermodynamic variable, but does not resolve the broader comparative question of which formulation is most advantageous.

References

Various formulations of the compressible Euler equations are currently being explored for the development of modern dynamical cores, differing in the choice of the thermodynamic variables, such as potential temperature, entropy, internal or total energy, as well as in the form of the momentum equation, e.g., the conservative or the vector-invariant form, and it is still unclear whether one formulation offers advantages over the others.

— Entropy-Stable and Well-Balanced Discontinuous Galerkin Methods for the Compressible Euler Equations in Vector-Invariant Form  (2610.05820 - Artiano et al., 5 Oct 2026) in Section 1, Introduction