Optimization of the Runge–Kutta iteration with residual smoothing

Iterative solvers in combination with multi-grid have been used extensively to solve large algebraic systems. One of the best known is the Runge–Kutta iteration. Previously (Haelterman et al. (2009) [3]) we reformulated the Runge–Kutta scheme and established a model of a complete V-cycle which was used to optimize the coefficients of the multi-stage scheme and resulted in a better overall performance. We now look into aspects of central and upwind residual smoothing within the same optimization framework. We consider explicit and implicit residual smoothing and either apply it within the Runge–Kutta time-steps, as a filter for restriction or as a preconditioner for the discretized equations. We also shed a different light on the very high CFL numbers obtained by upwind residual smoothing and point out that damping the high frequencies by residual smoothing is not necessarily a good idea.