p-Multigrid matrix-free discontinuous Galerkin solution strategies for the under-resolved simulation of incompressible turbulent flows

In recent years several research efforts focused on the development of high-order discontinuous Galerkin (dG) methods for scale resolving simulations of turbulent flows. Nevertheless, in the context of incompressible flow computations, the computational expense of solving large scale equation systems characterized by indefinite Jacobian matrices has often prevented from dealing with industrially-relevant computations. In this work we seek to improve the efficiency of Rosenbrock-type linearly-implicit Runge-Kutta methods by devising robust, scalable and memory-lean solution strategies. In particular, we introduce memory saving p-multigrid preconditioners coupling matrix-free and matrix-based Krylov iterative smoothers. The p-multigrid preconditioner relies on cheap block-diagonal smoother's preconditioners on the fine space to reduce assembly costs and memory allocation, and ensures an adequate resolution of the coarsest space of the multigrid iteration using Additive Schwarz precondioned smoothers to obtain satisfactory convergence rates and optimal parallel efficiency of the method. Extensive numerical validation is performed. The Rosenbrock formulation is applied to test cases of growing complexity: the laminar unsteady flow around a two-dimensional cylinder at Re=200 and around a sphere at Re=300, the transitional flow problem of the ERCOFTAC T3L test case suite with different levels of free-stream turbulence. As proof of concept, the numerical solution of the Boeing Rudimentary Landing Gear test case at Re=10^6 is reported. A good agreement of the solutions with experimental data is documented, as well as strong memory savings and execution time gains with respect to state-of-the art solution strategies.