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A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations

Andrew R. Winters, Rodrigo C. Moura, Gianmarco Mengaldo, Gregor J. Gassner, Stefanie Walch, Joaquim Peiro, Spencer J. Sherwin

arXiv:1711.10180v2math.NAcs.NA

TL;DR

Under-resolved, high-Reynolds-number turbulence challenges DG methods because nonlinear under-integration causes aliasing and stability problems. The paper compares consistent/over-integration and conservative split-form DGSEM for inviscid TGV computations. At moderately high orders their solution quality is similar, while stability and accuracy depend on order and flux choice.

  • Problem

    Under-resolved high-Reynolds-number turbulence poses accuracy and stability challenges for DG methods because small-scale dissipation may be insufficient and under-integrated nonlinear terms cause aliasing.

    Source: §1 · p. 2

  • Method

    The study compares consistent/over-integration with split-form DGSEM for under-resolved inviscid TGV computations, using Roe-type and local Lax-Friedrichs fluxes.

    Source: §1 · p. 3, §6 · p. 19

  • Results

    At moderately high orders, split forms perform very similarly to consistent integration, but at lower orders solution quality differs and at very high orders stable solutions can show unphysical features.

    Source: §6 · pp. 20–21

  • Takeaways & Limitations

    DG-based model-free turbulence computations are more likely to be stable and accurate with complete Riemann solvers such as Roe’s flux at moderately high orders.

    Source: §6 · p. 21

    Fully central split forms are stable only at low polynomial orders (N ≤3); the proposed physical-origin explanation for TGV instabilities remains speculative.

    Source: §4 · p. 14

Abstract

from arXiv · show

This work focuses on the accuracy and stability of high-order nodal discontinuous Galerkin (DG) methods for under-resolved turbulence computations. In particular we consider the inviscid Taylor-Green vortex (TGV) flow to analyse the implicit large eddy simulation (iLES) capabilities of DG methods at very high Reynolds numbers. The governing equations are discretised in two ways in order to suppress aliasing errors introduced into the discrete variational forms due to the under-integration of non-linear terms. The first, more straightforward way relies on consistent/over-integration, where quadrature accuracy is improved by using a larger number of integration points, consistent with the degree of the non-linearities. The second strategy, originally applied in the high-order finite difference community, relies on a split (or skew-symmetric) form of the governing equations. Different split forms are available depending on how the variables in the non-linear terms are grouped. The desired split form is then built by averaging conservative and non-conservative forms of the governing equations, although conservativity of the DG scheme is fully preserved. A preliminary analysis based on Burgers' turbulence in one spatial dimension is conducted and shows the potential of split forms in keeping the energy of higher-order polynomial modes close to the expected levels. This indicates that the favourable dealiasing properties observed from split-form approaches in more classical schemes seem to hold for DG. The remainder of the study considers a comprehensive set of (under-resolved) computations of the inviscid TGV flow and compares the accuracy and robustness of consistent/over-integration and split form discretisations based on the local Lax-Friedrichs and Roe-type Riemann solvers...

1. Introduction

Under-resolved high-Reynolds-number turbulence challenges spectral element methods because their dissipation may not stabilize small scales and under-integrated nonlinear terms introduce aliasing errors. This study compares over-integration and split-form DG discretisations for accuracy and robustness in inviscid TGV computations.

1. Introduction

  • Under-resolved high-Reynolds-number flows retain significant small-scale energy, while spectral element methods may provide insufficient dissipation to stabilize them.Under-integration of nonlinear terms also introduces aliasing-driven instabilities that can cause numerical divergence, especially at higher orders.Source: §1 · p. 2
  • Subgrid-scale models and spectral vanishing viscosity offer routes to small-scale stabilization, while DG-based iLES can rely on numerical dissipation instead of an explicit turbulence model.DG upwind dissipation can stabilize advection-dominated flows at moderate Reynolds numbers, but may be insufficient at high Reynolds numbers and higher orders, even with over-integration.Source: §1 · pp. 2–3
  • The study compares consistent/over-integration with split-form DG for inviscid TGV computations, assessing solution quality and robustness when viscous effects are negligible.Split forms average conservative and advective forms while preserving DG conservation; the study focuses on dealiasing issue (b).Source: §1 · p. 3

2. Discontinuous Galerkin discretisations

The section introduces nodal DGSEM and the two dealiasing strategies, then describes split forms and the interface fluxes used with them. Standard collocation is efficient but can under-integrate nonlinear fluxes; split forms use flux differencing to retain conservation.

2.1. Standard and over-integrated DGSEM approaches

  • Nodal DGSEM represents the solution with elementwise polynomial basis functions on a reference element, and numerical interface fluxes replace boundary terms in the weak formulation.The presentation restricts its discussion to straight-sided hexahedral elements and, for the simulations, Cartesian meshes.Source: §2.1 · pp. 3–4
  • Standard DGSEM collocates solution and quadrature at N+1 LGL nodes, an efficient choice that can under-integrate nonlinear fluxes and introduce aliasing errors.Over-integration uses more quadrature points; consistent integration selects quadrature accuracy according to the nonlinear fluxes, though compressible conservative-variable fluxes are rational and not exactly integrated with finite Q.Source: §2.1 · pp. 4–5

2.2. Split form DG discretisations

  • Split forms average conservative and advective forms of nonlinear terms; SBP flux differencing allows the considered DGSEM split forms to remain locally conservative.Symmetric, consistent sub-cell fluxes encode different split forms, while the SBP property mimics integration by parts discretely.Source: §2.2 · pp. 6–8

2.3. Selection of the numerical interface flux function

  • Over-integrated DGSEM uses classical LLF and Roe-type interface fluxes, while split-form DG couples its sub-cell flux choice to the interface flux.For the KG split with Roe flux, the dissipation term is modified to ensure kinetic-energy stability; the central interface flux lacks the usual dissipation term.Source: §2.3 · pp. 8–9

3. Analysis of polynomial aliasing errors of split form approaches

A frozen Burgers-turbulence analysis compares modal tendencies from conservative, non-conservative, and quadratic split DGSEM forms against exact integration. Conservative and non-conservative forms err in opposite directions at high modes, while the quadratic split balances these tendencies and better controls modal-energy variation.

3. Analysis of polynomial aliasing errors of split form approaches

  • The modal-space test uses a frozen solution whose modal energy follows a turbulent-like (j+1)^-5/3 scaling, and interprets its results by analogy with Fourier space.The authors acknowledge that this modal/Fourier analogy is limited.Source: §3 · p. 10
  • The frozen Burgers analysis uses a turbulent-like modal profile and compares transformed right-hand sides across polynomial modes against exact-integration reference values.The figure compares conservative, quadratic split, and non-conservative DGSEM forms at different polynomial orders.Source: §3 · pp. 10–11, Figure 1 · p. 11
  • Conservative DGSEM overpredicts modal-coefficient variation rates while non-conservative DGSEM underpredicts them, with the largest discrepancies at the highest modes.Those higher modes have larger relative modal-energy variation errors, and the resulting small-scale dynamics may be grossly miscalculated and induce numerical instabilities.Source: §3 · pp. 11–12
  • The quadratic split form predicts modal-energy variation more accurately by averaging the opposite tendencies of conservative and non-conservative DGSEM, acting as a built-in dealiasing mechanism.Its energy-variation levels tend to remain slightly below the correct levels.Source: §3 · p. 12
  • As quadrature points increase, conservative-form energy variation approaches exact values monotonically from above, while over-integration may still leave aliasing errors.The study notes prior evidence that appropriate split forms can address aliasing in advection problems with non-constant speed functions where over-integration cannot entirely eliminate it.Source: §3 · p. 12

4. TGV simulations and numerical stability

The inviscid TGV tests compare consistent integration with split forms across polynomial orders and examine how quadrature and flux choice affect stability. KG and DU split forms stabilized all tested cases, while some consistent-integration cases crashed.

4. TGV simulations and numerical stability

  • The study directly simulates inviscid compressible Euler equations (γ = 1.4) for a modified Taylor–Green vortex initial state, with reference scales ℓo = ρo = Vo = 1 and co = 10 (Mach 0.1), to T = 14.Source: §4 · p. 12
  • KG and DU split forms reached final time in every tested case, while some consistent-integration cases crashed.The cases are summarized in Table 1.Source: Table 1 · p. 13, §4 · p. 13
  • For m = 8 and nel = 14, LLF crash time increased with Q through 3m/2 and then was practically unchanged; Roe cases stabilized for Q ≥ 3m/2.The cases used consistent integration; Table 2 reports crash time against quadrature points.Source: §4 · p. 13, Table 2 · p. 14
  • The authors attribute LLF’s poorer stability relative to Roe likely to strong over-upwinding, which can increase the likelihood of TGV instabilities.Source: §4 · p. 14
  • The instability mechanism remains unresolved, and the proposed link between energy-conserving simulations and physical TGV singularities is speculative.Suitable split forms stabilize both Roe and LLF at high orders, whereas fully central, nearly kinetic-energy-conserving splits are stable only for N ≤ 3.Source: §4 · p. 14
  • Split forms have superior nonlinear stability, but their accuracy requires further investigation because they remain susceptible to aliasing errors.Source: §4 · p. 14

5. Solution quality for different dealiased DGSEM

At moderately high orders, KG split forms produce results close to consistent integration, while Roe-based flow profiles better match the canonical turbulent shape than LLF-based profiles. At very high orders and low orders, the comparisons reveal energy bumps and meaningful differences, respectively, with low-order QR profiles retaining a reasonably correct overall shape.

5. Solution quality for different dealiased DGSEM

  • The solution-quality assessment uses KG as the representative split because prior inviscid TGV results found similar flows among the most stable schemes, while KG offered simpler implementation and lower computational cost.The study also reports a brief KG–DU comparison in the appendix.Source: §5 · p. 15

5.1. Effect of the Riemann flux at moderately high-orders

  • At m = 5 and higher orders, KG spectra are close to consistent integration, but LLF-based cases show a spurious energy bump while Roe-based cases do not.At small scales, split-form spectra are slightly less energetic than consistent-integration spectra; the closeness also holds at higher orders, unlike for m ≤3.Source: §5.1 · pp. 15–16, Figure 2 · p. 15
  • Roe-based QR profiles closely resemble the canonical teardrop shape, whereas LLF-based profiles have a less pronounced tail and appear less physical.Split-form QR profiles are only slightly wider than those from consistent integration and are otherwise practically indistinguishable.Source: Figure 3 · p. 16, §5.1 · p. 17

5.2. Solution quality at lower and higher polynomial orders

  • At m = 16, consistent integration does not suppress TGV instabilities, and energy bumps can emerge even with Roe-type flux as polynomial order increases.The higher-order split-form solutions follow trends expected from consistent integration where those simulations remain stable.Source: §5.2 · p. 17, Figure 4 · p. 18
  • The very high-order Roe-based QR profile is symmetric and similar to the energy-preserving KG profile without interface dissipation, but both are affected by over-energetic bump-related scales.The energy-preserving KG solution without interface dissipation is stable for m ≤4; split forms without interface dissipation crash for m > 4.Source: §5.2 · p. 17, Figure 5 · p. 18
  • At m = 3, consistent-integration and split-form spectra differ, and LLF spectra retain notably high energy in small-scale regions during nearly homogeneous decay.The LLF spectra’s additional small-scale energy may reflect a mild energy bump, despite appearing bump-free initially.Source: §5.2 · pp. 17–18, Figure 6 · p. 19
  • At low orders, QR profiles differ between split forms and consistent integration; their overall shape remains reasonably correct, though the turbulent kinematics are less clean than at moderately higher orders with the same DOFs.The low-order profiles seem more physical than those from very-high-order discretisations.Source: §5.2 · pp. 18–19

6. Conclusions

At moderately high orders, split forms and consistent integration produce similar turbulent solutions, while very high-order solutions can show unphysical features. The study identifies Roe-type fluxes at moderately high orders as a promising choice for stable, accurate DG turbulence computations.

6. Conclusions

  • Both dealiasing strategies require extra computational effort, and their implementation optimization remains ongoing; curved-element geometric aliasing is left for future study.The conclusions recommend complete Riemann solvers such as Roe’s flux at moderately high orders for stable, accurate very-high-Reynolds-number computations.Source: §6 · pp. 19, 21
  • DGSEM split forms balance opposing high-mode energy errors in frozen one-dimensional Burgers’ turbulence, keeping energy slightly below correct levels and potentially helping control its growth.Conservative formulations tend to overestimate high-order-mode energy, whereas advective formulations tend to underestimate it.Source: §6 · p. 19
  • Relevant split forms are more robust than consistent integration in inviscid TGV tests, but lose stability when central fluxes replace a Riemann solver.The study argues that energy-conserving bias can make TGV discretisations prone to instability, though TGV instabilities remain incompletely understood.Source: §6 · pp. 19–20
  • At around sixth order, split forms and consistent integration produce very similar solution quality, with energy spectra and QR diagrams exhibiting expected physical trends.At third order or below, the approaches differ appreciably and both show inferior quality for the same number of degrees of freedom.Source: §6 · pp. 20–21
  • At eighth order or above, stable solutions show unphysical features attributed to an energy-conserving bias and sharper wavenumber-space dissipation.The bias is especially prominent with LLF flux, whose over-upwind character at low Mach numbers is identified as a contributor.Source: §6 · p. 21

Appendix A. Comparison between representative split forms

The appendix compares the DU and KG split forms for TGV flow and finds their reported diagnostics nearly indistinguishable in the examined case. KG is used as the representative form in later analyses because it is known to be more robust for properly compressible flows.

Appendix A. Comparison between representative split forms

  • For the m = 5, nel = 23 case with Roe’s flux, DU and KG yield practically indistinguishable enstrophy evolution and nearly identical energy spectra and QR diagrams.The QR result recovers the teardrop shape expected for turbulent flows; the spectrum is considered at t = 9, near peak kinetic-energy dissipation.Source: Appendix A · pp. 24–25, Figure A.8 · p. 25
  • The energy spectrum is sampled on an equispaced Cartesian grid that avoids elemental interfaces, using Ndof = (nel m)^3 points for each test case.The expected Kolmogorov -5/3 slope is referenced for the spectrum near peak kinetic-energy dissipation.Source: Appendix A · p. 24
  • KG is chosen as the representative split form because it is known to be more robust for properly compressible flows, although DU and KG may differ more when compressibility effects are stronger.The appendix notes that the QR diagrams were nearly identical, so only the KG diagram is shown alongside a pointwise comparison.Source: Appendix A · p. 25

Appendix B. Discussion of QR diagrams

QR diagrams represent joint probability distributions of the second and third invariants of the velocity-gradient tensor, providing a statistical view of possible local flow trajectories. Their normalized axes and zero-discriminant curve help interpret changes in turbulent kinematics.

Appendix B. Discussion of QR diagrams

  • QR diagrams map the joint PDFs of the second (Q) and third (R) invariants of the velocity-gradient tensor to represent possible linear local flow trajectories.The velocity-gradient tensor is decomposed into symmetric and antisymmetric parts, identified as the strain-rate and rotation-rate tensors, respectively.Source: Appendix B · p. 26
  • The diagrams normalize Q by ⟨SijSij⟩ and R by ⟨SijSij⟩^3/2; the white zero-discriminant curve separates topologically distinct flow patterns.Crossing that curve can be interpreted as a change in the flow trajectory of turbulent kinematics.Source: Appendix B · p. 26
  • The canonical turbulent teardrop profile resembles DNS diagrams of isotropic turbulence, with its tail aligned to the zero-discriminant curve for R > 0.The teardrop shape is described as a qualitatively universal feature across turbulent scenarios.Source: Appendix B · p. 26

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