Improving Euler computations at low Mach numbers

This paper consists of two parts, both dealing with conditioning techniques for low-Mach-number Euler-flow computations, in which a multigrid technique is applied. In the first part, for subsonic flows and upwind-discretized linearized 1-D Euler equations, the smoothing behavior of multigrid-accelerated point Gauss-Seidel relaxation is investigated. Error decay by convection over domain boundaries is also discussed. A fix to poor convergence rates at low Mach numbers is sought by 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. In the present case it is not necessary to solve this accuracy problem. In the second part, insight is given into the conditions of derivative matrices to be inverted in point-relaxation methods for 1-D and 2-D, upwind-discretized Euler equations. Speed regimes are found where ill-conditioning of these matrices occurs, and 1-D flow equations appear to be less well-conditioned than 2-D flow equations. Fixes to the ill-conditioning follow more or less directly, when thinking of adding regularizing matrices to the ill-conditioned derivative matrices. A smoothing analysis is made of point Gauss-Seidel relaxation applied to discrete Euler equations conditioned by such an additive matrix. The method is successfully applied to a very low-subsonic, steady, 2-D stagnation flow.