Optimized interface conditions for domain decomposition methods in fluid dynamics

Interface conditions (IC) between subdomains have an important impact on the convergence rate of domain decomposition algorithms. We first recall the Schwarz method which is based on the use of Dirichlet conditions on the boundaries of the subdomains and overlapping subdomains. We explain how it is possible to replace them by more efficient ICs with normal and tangential derivatives so that overlapping is not necessary. It is possible to optimize the coefficients of the IC in order to achieve the best convergence rate. Results are given for the convection-diffusion equation. Then we consider the compressible Euler equations which form a system of equations. We present a new analysis of the use of interface conditions based on the flux splitting. We compute the convergence rate in the Fourier space. We find a dependence of their effectiveness on the Mach number M. For M = 1/3, the convergence rate tends to zero as the wavenumber of the error goes to infinity. We stress the differences with the scalar equations. We present numerical results in agreement with the theoretical results.