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

Recasting an operator splitting solver into a standard finite volume  flux-based algorithm. The case of a Lagrange-Projection-type method for gas  dynamics