Non-modal analysis of spectral element methods: Towards accurate and robust large-eddy simulations

Fernández, Pablo; Moura, Rodrigo Costa; Mengaldo, Gianmarco; Peraire, J.

doi:10.1016/j.cma.2018.11.027

Cited by 39 publications

(31 citation statements)

References 84 publications

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“…Spectral element methods combine the advantageous properties of finite element/finite volume and spectral methods, namely geometric flexibility and reduced dispersion/dissipation errors (see e.g. [21,59,62] on the latter topic). Nevertheless, the application of spectral element methods to challenging problems such as turbulent flows over complex geometries is still somewhat hindered by numerical instability issues.…”

Section: Introductionmentioning

confidence: 99%

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

Winters

Moura

Mengaldo

et al. 2018

Journal of Computational Physics

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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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Section: Introductionmentioning

confidence: 99%

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

Winters

Moura

Mengaldo

et al. 2018

Journal of Computational Physics

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“…We note that, although a non-modal eigenanalysis strategy better suited for turbulence computations has been recently proposed [6], the present work will focus on more fundamental aspects and follow therefore the classical eigenanalysis. Finally, the results presented here are representative of a broader class of discontinuous SEM, given the well established connections within this class-see e.g.…”

Section: R C Moura ( )mentioning

confidence: 99%

“…where and ⊕ denote the left and right boundaries of element , in that order. As typical, expansions in (4) are to be inserted into (5)- (6), which are then required to hold for i = 0, . .…”

Section: Hdg Discretisationmentioning

confidence: 99%

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Viscous Diffusion Effects in the Eigenanalysis of (Hybridisable) DG Methods

Moura

Fernández

Mengaldo

et al. 2020

Lecture Notes in Computational Science and Engineering

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We present the first eigenanalysis of hybridisable discontinuous Galerkin (HDG) schemes for the advection-diffusion equation in one dimension. This study is also one of the first to include viscous diffusion effects in the eigenanalysis of discontinuous spectral element methods. The interplay between upwind dissipation and viscous diffusion is discussed and preliminary insights deemed relevant to (under-resolved) turbulence computation approaches are presented.

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“…In spite of the optimal convergence properties, the HDG methods presented in [88,108] might not be suitable for long-time computations, due to their energy-dissipative characteristics. The dissipative characteristics of HDG for convection-diffusion systems are investigated in [54]. Indeed, it has been observed that dissipative numerical schemes suffer a loss of accuracy for long-time computations, despite their optimal error estimates.…”

Section: Introductionmentioning

confidence: 99%

Hybridized Discontinuous Galerkin Methods for Wave Propagation

et al. 2018

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We present the recent development of hybridizable and embedded discontinuous Galerkin (DG) methods for wave propagation problems in fluids, solids, and electromagnetism. In each of these areas, we describe the methods, discuss their main features, display numerical results to illustrate their performance, and conclude with bibliography notes. The main ingredients in devising these DG methods are (i) a local Galerkin projection of the underlying partial differential equations at the element level onto spaces of polynomials of degree k to parametrize the numerical solution in terms of the numerical trace; (ii) a judicious choice of the numerical flux to provide stability and consistency; and (iii) a global jump condition that enforces the continuity of the numerical flux to obtain a global system in terms of the numerical trace. These DG methods are termed hybridized DG methods, because they are amenable to hybridization (static condensation) and hence to more efficient implementations. They share many common advantages of DG methods and possess some unique features that make them well-suited to wave propagation problems. *

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Non-modal analysis of spectral element methods: Towards accurate and robust large-eddy simulations

Cited by 39 publications

References 84 publications

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

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

Viscous Diffusion Effects in the Eigenanalysis of (Hybridisable) DG Methods

Hybridized Discontinuous Galerkin Methods for Wave Propagation

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