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 Peclet 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. (c) 2022 The Authors. Published by Elsevier Inc.

Automated summary

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, including the number of coarse-level iterations, the polynomial order, and the combination of h- and p-multigrid, is explored to find the most efficient configurations. V-cycle p-multigrid is shown to be more efficient at higher polynomial orders, and the use of W-cycles and/or hp-multigrid appears to offer additional advantages. Additionally, the influence of high Peclet numbers and high aspect-ratio cells on error dissipation is investigated in the 2D case. Finally, the non-modal dissipation is related to the convergence rate of the multigrid through a series of manufactured solutions.