For subsonic flows and upwind-discretized, linearized 1-D Euler equations, the smoothing behavior of multigrid-accelerated point Gauss-Seidel relaxation is analyzed. Error decay by convection across domain boundaries is also discussed. A fix to poor convergence rates at low Mach numbers is sought in replacing the point relaxation applied to unconditioned Euler equations, by locally implicit "time"-stepping applied to preconditioned Euler equations. The locally implicit iteration step is optimized for good damping of high-frequency errors. Numerical inaccuracy at low Mach numbers is also addressed.
Koren, B., & Leer, van, B. (1995). Analysis of preconditioning and multigrid for Euler flows with low-subsonic regions. Advances in Computational Mathematics, 4(1), 127-144. https://doi.org/10.1007/BF02123476