Domain decomposition algorithms for the compressible Euler equations

In this work we present an overview of some classical and new domain decomposition methods for the resolution of the Euler equations. The classical Schwarz methods are formulated and analyzed in the framework of first order hyperbolic systems and the differences with respect to the scalar problems are presented. This kind of algorithms behave quite well for bigger Mach numbers but we can further improve their performances in the case of lower Mach numbers. There are two possible ways to achieve this goal. The first one implies the use of the optimized interface conditions depending on a few parameters that generalize the classical ones. The second is inspired from the Robin-Robin preconditioner for the convection-diffusion equation by using the equivalence via the Smith factorization with a third order scalar equation.