  In this work we design a new domain decomposition method for the Euler
equations in 2 dimensions. The basis is the equivalence via the Smith
factorization with a third order scalar equation to whom we can apply an
algorithm inspired from the Robin-Robin preconditioner for the
convection-diffusion equation. Afterwards we translate it into an algorithm for
the initial system and prove that at the continuous level and for a
decomposition into 2 sub-domains, it converges in 2 iterations. This property
cannot be preserved strictly at discrete level and for arbitrary domain
decompositions but we still have numerical results which confirm a very good
stability with respect to the various parameters of the problem (mesh size,
Mach number, ....).
