Convergence characteristics of preconditioned Euler equations

The convergence characteristics of the preconditioned Euler equations were studied. A perturbation analysis was conducted to search for the relationships between the convergence characteristics and the flow Mach numbers. Also, the influence of cancellation errors on the convergence characteristics was investigated. The governing equations were the preconditioned two-dimensional Euler equations. Flows in a two-dimensional channel with a 10% circular bump in the middle of the channel were calculated at different speeds. Roe’s FDS scheme was used for spatial discretization and the LU-SGS (Lower Upper Symmetric Gauss Seidel) scheme was used for time integration. It was shown that the convergence characteristics of continuity and momentum equations were maintained regardless of the Mach numbers, but the convergence characteristics of energy equation were strongly dependant on the Mach numbers and worsened as the Mach number decreased. The convergence characteristics were well explained by the perturbation analysis. The convergence characteristics were strongly dependant on the characteristics of the preconditioning matrix. Cancellation errors caused a serious convergence problem especially in the calculation of the energy equation at very low Mach numbers. The relative treatments and a higher precision of floating-point variables alleviated cancellation problems to some extent. However, the convergence rate was not affected by the relative treatments or by the precision of floating-point variables.