  In this paper, we propose a modification of an acoustic-transport operator
splitting Lagrange-projection method for simulating compressible flows with
gravity. The original method involves two steps that respectively account for
acoustic and transport effects. Our work proposes a simple modification of the
transport step, and the resulting modified scheme turns out to be a
flux-splitting method. This new numerical method is less computationally
expensive, more memory efficient, and easier to implement than the original
one. We prove stability properties for this new scheme by showing that under
classical CFL conditions, the method is positivity preserving for mass, energy
and entropy satisfying. The flexible flux-splitting structure of the method
enables straightforward extensions of the method to multi-dimensional problems
(with respect to space) and high-order discretizations that are presented in
this work. We also propose an interpretation of the flux-splitting solver as a
relaxation approximation. Both the stability and the accuracy of the new method
are tested against one-dimensional and two-dimensional numerical experiments
that involve highly compressible flows and low-Mach regimes.
