Efficient discontinuous Galerkin implementations and preconditioners for implicit unsteady compressible flow simulations. (15th May 2020) - Franciolini, Matteo; Fidkowski, Krzysztof J.; Crivellini, Andrea
Highlights: Novel approximate-inherited multigrid improve efficiency of HDG. Novel primal HDG approach reduce matrix assembly time of HDG. Multigrid matrix-free reduce memory footprint of DG. Primal HDG excels for stiff discretizations/large time steps. Multigrid matrix-free convenient with physics-constrained time steps. Abstract: This work presents and compares efficient implementations of high-order discontinuous Galerkin methods: a modal matrix-free discontinuous Galerkin (DG) method, a hybridizable discontinuous Galerkin (HDG) method, and a primal formulation of HDG, applied to the implicit solution of unsteady compressible flows. The matrix-free implementation allows for a reduction of the memory footprint of the solver when dealing with implicit time-accurate discretizations. HDG reduces the number of globally-coupled degrees of freedom relative to DG, at high order, by statically condensing element-interior degrees of freedom from the system in favor of face unknowns. The primal formulation further reduces the element-interior degrees of freedom by eliminating the gradient as a separate unknown. This paper introduces a p -multigrid preconditioner implementation for these discretizations and presents results for various flow problems. Benefits of the p -multigrid strategy relative to simpler, less expensive, preconditioners are observed for stiff systems, such as those arising from low-Mach number flows at high-order approximation. The p -multigrid preconditioner also shows excellent scalability for parallel computations. Additional savings in both speed and memory occur with a matrix-free/reduced version of the preconditioner.