Non-modal analysis of multigrid schemes for the high-order flux reconstruction method

The present study introduces an application of the non-modal analysis to multigrid operators with explicit Runge-Kutta smoothers in the context of Flux Reconstruction discretizations of the linear convection-diffusion equation. A dissipation curve is obtained that reflects upon the convergence properties of the multigrid operator. The number of smoothing steps, the type of cycle (V/W) and the combination of p-and h-multigrid are taken into account in order to find those configurations which yield faster convergence rates. The analysis is carried out for polynomial orders up to P = 6, in 1D and 2D for varying degrees of convection (Péclet number), as well as for high aspect ratio cells. The non-modal analysis can support existing evidence on the behaviour of multigrid schemes. W-cycles, a higher number of coarse-level sweeps or the combined use of hp-multigrid are shown to increase the error dissipation, while higher degrees of convection and/or high aspect-ratio cells both decrease the error dissipation rate.