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.