Multi-stage solvers optimized for damping and propagation

Explicit multi‐stage solvers are routinely used to solve the semi‐discretized equations that arise in Computational Fluid Dynamics (CFD) problems. Often they are used in combination with multi‐grid methods. In that case, the role of the multi‐stage solver is to efficiently reduce the high frequency modes on the current grid and is called a smoother. In the past, when optimizing the coefficients of the scheme, only the damping characteristics of the smoother were taken into account and the interaction with the remainder of the multi‐grid cycle was neglected. Recently it had been found that coefficients that result in less damping, but allow for a higher Courant–Friedrichs–Lewy (CFL) number are often superior to schemes that try to optimize damping alone. While this is certainly true for multi‐stage schemes used as a stand‐alone solver, we investigate in this paper if using higher CFL numbers also yields better results in a multi‐grid setting. We compare the results with a previous study we conducted and where a more accurate model of the multi‐grid cycle was used to optimize the various parameters of the solver. We show that the use of the more accurate model results in better coefficients and that in a multi‐grid setting propagation is of little importance. We also look into the gains to be made when we allow the parameters to be different for the pre‐ and post‐smoother and show that even better coefficients can be found in this way.