This work addresses the simulation of compressible flows on unstructured meshes across a wide range of Mach numbers. To this end, we propose a staggered finite volume method for the compressible Euler equations in both barotropic and full formulations. Over the staggered grid, cell-centered unknowns include density, pressure, and energy, while face-centered variables represent velocity related quantities.The method is based on a Lagrange-projection like acoustic-transport splitting combined with a Suliciu-type relaxation approximation for the acoustic subsystem.Internal and kinetic energies are discretized on distinct sets of degrees of freedom, and a corrective mechanism is introduced to ensure consistency with the total energy conservation in the full Euler case.The proposed approach preserves the conservation properties of the compressible Euler equations, remains applicable to general equations of state, and satisfies key stability conditions: positivity of density (for barotropic and full systems), positivity of internal energy (for the full system), compliance with the second law of thermodynamics (globally for the barotropic system and both globally and locally for the full system), and local conservation of the total energy on the dual mesh for the full Euler equations.The time step is constrained only by the material velocity, and each time update requires solving a single linear system for the pressure.The low-Mach-number behavior of the scheme is analyzed, and the preservation of constant pressure--velocity profiles is established for stiffened-gas equations of state.The performance and robustness of the method are illustrated through numerical experiments for both barotropic and full Euler equations, in one and two space dimensions, on structured and unstructured meshes, over a wide range of Mach numbers including the low-Mach regime.
Halabi et al. (Wed,) studied this question.