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

Winters, Andrew R. and Moura, Rodrigo C. and Mengaldo, Gianmarco and Gassner, Gregor J. and Walch, Stefanie and Peiro, Joaquim and Sherwin, Spencer J. (2018) A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations. Journal of Computational Physics, 372 . pp. 1-21. ISSN 0021-9991.

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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. Recent works showed that relevant split forms can stabilize higher-order inviscid TGV test cases otherwise unstable even with consistent integration. Here we show that stable high-order cases achievable with both strategies have comparable accuracy, further supporting the good dealiasing properties of split form DG. The higher-order cases achieved only with split form schemes also displayed all the main features expected from consistent/over-integration. Among test cases with the same number of degrees of freedom, best solution quality is obtained with Roe-type fluxes at moderately high orders (around sixth order). Solutions obtained with very high polynomial orders displayed spurious features attributed to a sharper dissipation in wavenumber space. Accuracy differences between the two dealiasing strategies considered were, however, observed for the low-order cases, which also yielded reduced solution quality compared to high-order results.

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Additional Information:© 2018 The Author(s). Published by Elsevier Inc. Under a Creative Commons license. Attribution 4.0 International (CC BY 4.0) Open Access funded by Engineering and Physical Sciences Research Council. Received 28 November 2017, Revised 17 May 2018, Accepted 4 June 2018, Available online 18 June 2018. Andrew Winters was partially supported by the “Mobility Grant for National and International Young Faculty” from the University of Cologne. Stefanie Walch thanks the Deutsche Forschungsgemeinschaft (DFG) for funding through the SPP 1573 “The physics of the interstellar medium” and the funding from the European Research Council via the ERC Starting Grant “The radiative interstellar medium” (RADFEEDBACK). Gregor Gassner has been supported by the European Research Council (ERC) under the European Union's Eights Framework Program Horizon 2020 with the research project Extreme, ERC grant agreement no. 714487. Rodrigo Moura would like to acknowledge funding under the Brazilian Science without Borders scheme. Spencer Sherwin and Joaquim Peiró acknowledge support from the Engineering and Physical Sciences Research Council (EPSRC) under grant EP/L000407/1. Spencer Sherwin additionally acknowledges support as Royal Academy of Engineering Research Chair under grant 10145/86. The simulations in this work were partially performed using the Cologne High Efficiency Operating Platform for Sciences (CHEOPS) HPC cluster at the Regionales Rechenzentrum Köln (RRZK), University of Cologne, Germany. Use of the HPC facilities at Imperial College London is also acknowledged.
Funding AgencyGrant Number
University of CologneUNSPECIFIED
Deutsche Forschungsgemeinschaft (DFG)SPP 1573
European Research Council (ERC)679852 RADFEEDBACK
European Research Council (ERC)714487
Brazilian Science without BordersUNSPECIFIED
Engineering and Physical Sciences Research Council (EPSRC)EP/L000407/1
Royal Academy of Engineering Research Chair10145/86
Subject Keywords:Spectral element methods; Discontinuous Galerkin; Polynomial dealiasing; Split form schemes; Implicit large eddy simulation; Inviscid Taylor-Green vortex
Record Number:CaltechAUTHORS:20180619-101211192
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Official Citation:Andrew R. Winters, Rodrigo C. Moura, Gianmarco Mengaldo, Gregor J. Gassner, Stefanie Walch, Joaquim Peiro, Spencer J. Sherwin, A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations, Journal of Computational Physics, Volume 372, 2018, Pages 1-21, ISSN 0021-9991, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:87219
Deposited By: Tony Diaz
Deposited On:19 Jun 2018 17:21
Last Modified:06 Jul 2018 15:50

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