Non-modal analysis of linear multigrid schemes for the high-order Flux Reconstruction method

We present a numerical analysis of linear multigrid operators for the high-order Flux Reconstruction method. The non-modal analysis is used to assess the short-term numerical dissipation in the context of 1D and 2D linear convection-diffusion. The effect of several parameters, namely the number of coarse-level iterations, the polynomial order and the combination of h- and p-multigrid is explored in an effort to identify the most efficient configurations. V-cycle p-multigrid is shown to have increased efficiency at higher polynomial orders, and the use of W-cycles and/or hp-multigrid appear to offer additional advantages. The effect of high Péclet numbers and high aspect-ratio cells is also explored in 2D, and both factors are shown to decrease the error dissipation. Finally, we relate the non-modal dissipation to the convergence rate of the multigrid in a series of manufactured solutions.