High-order DG solvers for under-resolved turbulent incompressible flows: A comparison of L2L^2 and HH(div) methods

The accurate numerical simulation of turbulent incompressible flows is a challenging topic in computational fluid dynamics. For discretisation methods to be robust in the under-resolved regime, mass conservation as well as energy stability are key ingredients to obtain robust and accurate discretisations. Recently, two approaches have been proposed in the context of high-order discontinuous Galerkin (DG) discretisations that address these aspects differently. On the one hand, standard $L^2$-based DG discretisations enforce mass conservation and energy stability weakly by the use of additional stabilisation terms. On the other hand, pointwise divergence-free $H(\operatorname{div})$-conforming approaches ensure exact mass conservation and energy stability by the use of tailored finite element function spaces. The present work raises the question whether and to which extent these two approaches are equivalent when applied to under-resolved turbulent flows. This comparative study highlights similarities and differences of these two approaches. The numerical results emphasise that both discretisation strategies are promising for under-resolved simulations of turbulent flows due to their inherent dissipation mechanisms.Comment: 24 pages, 13 figure

An embedded-hybridized discontinuous Galerkin method for the coupled Stokes-Darcy system

We introduce an embedded-hybridized discontinuous Galerkin (EDG-HDG) method for the coupled Stokes-Darcy system. This EDG-HDG method is a pointwise mass-conserving discretization resulting in a divergence-conforming velocity field on the whole domain. In the proposed scheme, coupling between the Stokes and Darcy domains is achieved naturally through the EDG-HDG facet variables. \emph{A priori} error analysis shows optimal convergence rates, and that the velocity error does not depend on the pressure. The error analysis is verified through numerical examples on unstructured grids for different orders of polynomial approximation

h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

In this work we exploit agglomeration based $h$-multigrid preconditioners to speed-up the iterative solution of discontinuous Galerkin discretizations of the Stokes and Navier-Stokes equations. As a distinctive feature $h$-coarsened mesh sequences are generated by recursive agglomeration of a fine grid, admitting arbitrarily unstructured grids of complex domains, and agglomeration based discontinuous Galerkin discretizations are employed to deal with agglomerated elements of coarse levels. Both the expense of building coarse grid operators and the performance of the resulting multigrid iteration are investigated. For the sake of efficiency coarse grid operators are inherited through element-by-element $L^2$ projections, avoiding the cost of numerical integration over agglomerated elements. Specific care is devoted to the projection of viscous terms discretized by means of the BR2 dG method. We demonstrate that enforcing the correct amount of stabilization on coarse grids levels is mandatory for achieving uniform convergence with respect to the number of levels. The numerical solution of steady and unsteady, linear and non-linear problems is considered tackling challenging 2D test cases and 3D real life computations on parallel architectures. Significant execution time gains are documented.Comment: 78 pages, 7 figure

Pseudo-transient Continuation, Solution Update Methods, and CFL Strategies for DG Discretizations of the RANS-SA Equations

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/106459/1/AIAA2013-2686.pd