Lagrange-Flux schemes and the entropy property

The Lagrange-Flux schemes are Eulerian finite volume schemes that make use of an approximate Riemann solver in Lagrangian description with particular upwind convective fluxes. They have been recently designed as variant formulations of Lagrange-remap schemes that provide better HPC performance on modern multicore processors, see~[De Vuyst et al., OGST 71(6), 2016]. Actually Lagrange-Flux schemes show several advantages compared to Lagrange-remap schemes, especially for multidimensional problems: they do not require the computation of deformed Lagrangian cells or mesh intersections as in the remapping process. The paper focuses on the entropy property of Lagrange-Flux schemes in their semi-discrete in space form, for one-dimensional problems and for the compressible Euler equations as example. We provide pseudo-viscosity pressure terms that ensure entropy production of order $O(|\Delta u|^3)$, where $|\Delta u|$ represents a velocity jump at a cell interface. Pseudo-viscosity terms are also designed to vanish into expansion regions as it is the case for rarefaction waves.