A short note on the discontinuous Galerkin discretization of the pressure projection operator in incompressible flow

Steinmoeller, Derek; Stastna, Marek; Lamb, Kevin G.

doi:10.1016/j.jcp.2013.05.036

Cited by 21 publications

(59 citation statements)

References 20 publications

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“…Additionally, we note that very recently [34] reported a similar type of instability as presented in this work, which was attributed to the use of a non-conformal DG formulation. However, in the present work we show that these instabilities are inherent to the temporal scheme and independent of the spatial discretisation (and the continuity/discotninuity of the underlying spaces).…”

Section: Introductionsupporting

confidence: 83%

Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations

Ferrer

Moxey

Willden

et al. 2014

Commun. comput. phys.

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Abstract. This paper presents limits for stability of projection type schemes when using high order pressure-velocity pairs of same degree. Two high order h/p variational methods encompassing continuous and discontinuous Galerkin formulations are used to explain previously observed lower limits on the time step for projection type schemes to be stable [18], when h-or p-refinement strategies are considered. In addition, the analysis included in this work shows that these stability limits do not depend only on the time step but on the product of the latter and the kinematic viscosity, which is of particular importance in the study of high Reynolds number flows. We show that high order methods prove advantageous in stabilising the simulations when small time steps and low kinematic viscosities are used.Drawing upon this analysis, we demonstrate how the effects of this instability can be reduced in the discontinuous scheme by introducing a stabilisation term into the global system. Finally, we show that these lower limits are compatible with CourantFriedrichs-Lewy (CFL) type restrictions, given that a sufficiently high polynomial order or a mall enough mesh spacing is selected.

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Section: Introductionsupporting

confidence: 83%

Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations

Ferrer

Moxey

Willden

et al. 2014

Commun. comput. phys.

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“…We analyze the efficiency of the solution of linear systems of equations in terms of the number of iterations required to solve the pressure Poisson equation (31), the projection equation (32), and the Helmholtz-like equation (33). Results are presented in Table 2.…”

Section: Efficiency Of Solvers and Preconditionersmentioning

confidence: 99%

“…The Helmholtz-like equation and the projection equation are preconditioned by the inverse mass matrix operator. In the following, we analyze the performance of the inverse mass matrix operator in addition to the three operators constituting the linear systems of equations (31), (32), and (33).…”

Section: Efficiency Of Matrix-free Implementationmentioning

confidence: 99%

Efficiency of high‐performance discontinuous Galerkin spectral element methods for under‐resolved turbulent incompressible flows

Fehn

Wall

Kronbichler

2018

Numerical Methods in Fluids

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Summary The present paper addresses the numerical solution of turbulent flows with high‐order discontinuous Galerkin methods for discretizing the incompressible Navier‐Stokes equations. The efficiency of high‐order methods when applied to under‐resolved problems is an open issue in the literature. This topic is carefully investigated in the present work by the example of the three‐dimensional Taylor‐Green vortex problem. Our implementation is based on a generic high‐performance framework for matrix‐free evaluation of finite element operators with one of the best realizations currently known. We present a methodology to systematically analyze the efficiency of the incompressible Navier‐Stokes solver for high polynomial degrees. Due to the absence of optimal rates of convergence in the under‐resolved regime, our results reveal that demonstrating improved efficiency of high‐order methods is a challenging task and that optimal computational complexity of solvers and preconditioners as well as matrix‐free implementations are necessary ingredients in achieving the goal of better solution quality at the same computational costs already for a geometrically simple problem such as the Taylor‐Green vortex. Although the analysis is performed for a Cartesian geometry, our approach is generic and can be applied to arbitrary geometries. We present excellent performance numbers on modern cache‐based computer architectures achieving a throughput for operator evaluation of 3·108 up to 1·109 DoFs/s (degrees of freedom per second) on one Intel Haswell node with 28 cores. Compared to performance results published within the last five years for high‐order discontinuous Galerkin discretizations of the compressible Navier‐Stokes equations, our approach reduces computational costs by more than one order of magnitude for the same setup.

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“…For all K ∈  h and ∈ L 2 ( K), the discrete (H 1 ) lifting ( ) ∈ V h | K ∩ L 2 0 (K) is defined as Note that the space for the discrete lifting is chosen such that (·) is unique and allows to rewrite in the definition of a h (·, ·). Our H(div)-HDG method with (H 1 ) lifting results from adding the form to the left-hand side of (14). One easily checks that this additional term ensures nonnegativity of the bilinear form a h (·, ·) for any ⩾ 1.…”

Section: Different Viscosity Treatment For the H(div)-hdg Methodsmentioning

confidence: 99%

“…While DG methods have reached a mature state in the field of the compressible Navier-Stokes equations 1-7 in the context of lowand moderate-Mach number turbulent flows, designing robust DG discretisations of the incompressible Navier-Stokes equations exhibits some subtleties in the context of underresolved turbulence: Applying well-known numerical flux formulations to the discretisation of convective and viscous terms only [8][9][10][11][12][13] has been realised to be not robust enough in underresolved scenarios, as pointed out in recent works. [14][15][16][17] It is worth mentioning that this lack of robustness is not related to underintegration of nonlinear terms, commonly known as aliasing. Instead, additional techniques are required, which are inherently linked to the nature of the incompressible Navier-Stokes equations and, in particular, the incompressibility constraint.…”

mentioning

confidence: 99%

High‐order DG solvers for underresolved turbulent incompressible flows: A comparison of L² and H(div) methods

Fehn

Kronbichler

Lehrenfeld

et al. 2019

Numerical Methods in Fluids

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Summary The accurate numerical simulation of turbulent incompressible flows is a challenging topic in computational fluid dynamics. For discretisation methods to be robust in the underresolved regime, mass conservation and 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 L2‐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(div)‐conforming approaches ensure exact mass conservation and energy stability by the use of tailored finite element function spaces. This work raises the question whether and to which extent these two approaches are equivalent when applied to underresolved turbulent flows. This comparative study highlights similarities and differences of these two approaches. The numerical results emphasise that both discretisation strategies are promising for underresolved simulations of turbulent flows due to their inherent dissipation mechanisms.

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A short note on the discontinuous Galerkin discretization of the pressure projection operator in incompressible flow

Cited by 21 publications

References 20 publications

Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations

Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations

Efficiency of high‐performance discontinuous Galerkin spectral element methods for under‐resolved turbulent incompressible flows

High‐order DG solvers for underresolved turbulent incompressible flows: A comparison of L² and H(div) methods

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A short note on the discontinuous Galerkin discretization of the pressure projection operator in incompressible flow

Cited by 21 publications

References 20 publications

Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations

Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations

Efficiency of high‐performance discontinuous Galerkin spectral element methods for under‐resolved turbulent incompressible flows

High‐order DG solvers for underresolved turbulent incompressible flows: A comparison of L2 and H(div) methods

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High‐order DG solvers for underresolved turbulent incompressible flows: A comparison of L² and H(div) methods