Super-Convergence  of Discontinuous Galerkin Method Applied to the Navier-Stokes Equations

Atkins, Harold L.

doi:10.2514/6.2009-3787

Cited by 8 publications

(6 citation statements)

References 20 publications

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“…This difference of one order can be observed in the definitions of the 'order of accuracy' proposed, somewhat heuristically, in [11,36]. It is clear now that those definitions correspond to the long-time rate obtained when h → 0 for constant k. Such long-time super-convergence is realized for problems exhibiting an eventual quasi-steady solution which does not depend on the initial condition [42]. However, for most other practical simulations, the realized convergence rate would typically lie somewhere in between the shorttime and long-time rates depending on the characteristic time-scale of the problem.…”

Section: Rate Of Convergence Of Pointwise Errormentioning

confidence: 94%

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On consistency and rate of convergence of Flux Reconstruction for time-dependent problems

Asthana

Watkins

Jameson

2017

Journal of Computational Physics

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This study is directed at a rigorous characterization of the consistency and convergence of discontinuous finite element schemes formulated using Flux Reconstruction (FR). We show that the FR formulation is consistent for linear advection and converges to the exact solution for any scheme that is stable in the von Neumann sense. The numerical solution for a scheme of polynomial order P is composed of P + 1 eignemodes, of which, one and exactly one is 'physical' such that it exhibits the analytical dispersion behavior in the limit of asymptotic grid resolution. The remaining P modes are 'spurious' such that the fraction of energy received by them from the initial condition vanishes in the asymptotic limit. On grid refinement, the rate of convergence of the numerical solution is a function of time, starting from a short-time rate at t = 0 + , associated with interpolation, and asymptotically approaching a long-time rate as t → ∞, associated with numerical differentiation. Both these rates can be inferred directly from the eigensystem of the numerical derivative operator. We verify these analytical expectations using simple experiments in 1-D and 2-D.

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Section: Rate Of Convergence Of Pointwise Errormentioning

confidence: 94%

“…In regard to applications, such a rate is relevant only for problems that either have periodic boundary conditions or exhibit a quasi-steady solution of interest [42].…”

Section: Linear Advection-diffusion In 2-dmentioning

confidence: 99%

On consistency and rate of convergence of Flux Reconstruction for time-dependent problems

Asthana

Watkins

Jameson

2017

Journal of Computational Physics

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“…3 ) cos(πx) sin(πŷ) cos(πẑ) sin(θ) cos(πx) cos(πŷ) sin(πẑ)      (13) such that the steady compressible continuity equation is exactly satisfied. The pressure relation given by Eq.…”

Section: B Basline Results and Comparisons To Incompressible Solutionmentioning

confidence: 95%

“…12 The method has demonstrated superconvergence for the shedding frequency for viscous flow over a two-dimensional cylinder. 13 Section III. of this paper will present the Navier-Stokes equations and describe the formulation and implementation of the QFDG method for viscous flows.…”

Section: Introductionmentioning

confidence: 99%

Simulations of Compressible Taylor-Green Flow by a Discontinuous Galerkin Method

Atkins

2016

46th AIAA Fluid Dynamics Conference

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A high-order discontinuous Galerkin method is applied to the compressible Taylor-Green vortex at Mach 0.1 to evaluate the method, and to investigate alternative means of evaluating solution accuracy when an exact solution is not available. Two distinct variants of the viscous terms produced nearly identical results, which is consistent with the convection dominant nature of the flow. Grid refinements on hexahedral grids were performed to an effective grid resolution of 1042 3 degrees of freedom. Error analysis using norms of the change in the solution between grid refinements indicate that the solution is entering an asymptotic regime at an effective grid resolution of ≈369 3 degrees of freedom. Simulations on tetrahedral grids produced similar or better results, in particular, converging to peak enstrophy on much coarser grids. These well-resolved results indicate that the compressible solution is measurably different from the incompressible case, and that the incompressible case cannot be used to quantitatively evaluate the accuracy of a high-order method. Further, a volume-averaged quantity hides much of the details of the flow and is not a suitable metric for evaluating a high-order method. Norms based on direct comparison of a sampling of field values between successive grids provides a reliable measure of error convergence.

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“…Prior work validating a two-dimensional variant of this code demonstrated super-convergence of unsteady viscous flow in the wake of a cylinder. 4 Some recent work to improve robustness of the DG method in the presence of shocks is described in Ref. 5.…”

Section: Introductionmentioning

confidence: 99%

High-Order Simulation of Induced Disturbance in a Mach 6 Boundary Layer

Atkins

2013

43rd Fluid Dynamics Conference

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Simulations of the early evolution of induced unstable disturbances in a Mach 6 hypersonic boundary layer are presented for the purposes of validating a high-order discontinuous Galerkin Navier-Stokes simulation code. The simulations are modeled after experiments performed in the Boeing/AFOSR Mach-6 Quiet Tunnel at Purdue University. In these experiments, a forcing mechanism was employed to induce reproducible disturbances in a hypersonic boundary layer providing for the controlled study of the growth and breakdown of these disturbances into turbulent spots. Simulations revealed that the form and strength of the excitation can greatly influence the growth of the disturbance. In particular, at large forcing amplitude, the simulated forcing produces large advecting transients that appear to enhance the growth of the wave packet, relative to that of low amplitude forcing. Simulation results with large amplitude forcing agree well with experimental results while results from low amplitude forcing agree with linear stability theory.

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Super-Convergence of Discontinuous Galerkin Method Applied to the Navier-Stokes Equations

Cited by 8 publications

References 20 publications

On consistency and rate of convergence of Flux Reconstruction for time-dependent problems

On consistency and rate of convergence of Flux Reconstruction for time-dependent problems

Simulations of Compressible Taylor-Green Flow by a Discontinuous Galerkin Method

High-Order Simulation of Induced Disturbance in a Mach 6 Boundary Layer

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