Revisiting the spectral analysis for high-order spectral discontinuous methods

Vanharen, Julien; Puigt, Guillaume; Vasseur, Xavier; Boussuge, Jean‐François; Sagaut, Pierre

doi:10.1016/j.jcp.2017.02.043

Cited by 43 publications

(45 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The dissipation of a certain scheme can be defined as the loss of energy of the initial wave, while the dispersion is the phase shift between the exact and numerical wave solutions [29]. Considering the numerical and exact solution polynomials resulted from the semi-discrete Fourier analysis, Eqs.…”

Section: True Behavior Of Dg Schemes Through a Combined-mode Semi-dismentioning

confidence: 99%

“…In practice, implicit LES (ILES) has been shown to perform very well for a variety of flow problems [13,15,16,17,18,19]. 25 In order to assess the dispersion/dissipation characteristics and resolution of a numerical scheme, Fourier analysis [20] is often utilized either in a semi-discrete [21,22,23,24,25] or fully discrete setting [26,27,28,29]. In our present work, we start with a review of semi-discrete schemes, and then proceed to analyze the fully discrete schemes assuming a periodic boundary condition.…”

Section: Introductionmentioning

confidence: 99%

“…In their work, these modes are replicates of the physical-mode along the wavenumber axis and they improve the accuracy of the scheme. Vanharen et al [29] concluded that after a large number of iterations, high-order schemes behave in dispersion and dissipation according to the 35 physical-mode asymptotically, for wavenumbers less than π. Nevertheless, the complete behavior of DG-type high-order schemes in dispersion and dissipation based on all eigenmodes has not been studied before.…”

mentioning

confidence: 99%

“…1. On the other hand, Vanharen et al [29] utilized the matrix power method to identify the asymptotic behavior of fullydiscrete SD schemes coupled with RK time integration schemes for kh ≤ π, and proposed new definitions of dispersion/dissipation for wavenumbers kh > π.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Fourier analysis and evaluation of DG, FD and compact difference methods for conservation laws

Alhawwary

Wang

2018

Journal of Computational Physics

View full text Add to dashboard Cite

Large eddy simulation (LES) has been increasingly used to tackle vortex-dominated turbulent flows. In LES, the quality of the simulation results hinges upon the quality of the numerical discretizations in both space and time. It is in this context we perform a Fourier analysis of several popular methods in LES including the discontinuous Galerkin (DG), finite difference (FD), and compact difference (CD) methods. We begin by reviewing the semi-discrete versions of all methods under-consideration, followed by a fully-discrete analysis with explicit Runge-Kutta (RK) time integration schemes. In this regard, we are able to unravel the true dispersion/dissipation behavior of DG and Runge-Kutta DG (RKDG) schemes for the entire wavenumber range. The physical-mode is verified to be a good approximation for the asymptotic behavior of these DG schemes in the low wavenumber range. After that, we proceed to compare the DG, FD, and CD methods in dispersion and dissipation properties. Numerical tests are conducted using the linear advection equation to verify the analysis. In comparing different methods, it is found that the overall numerical dissipation strongly depends on the time step. Compact difference (CD) and central finite difference (FD) schemes, in some particular settings, can have more numerical dissipation than the DG scheme with an upwind flux. This claim is then verified through a numerical test using the Burgers' equation. 30 22, 24] defined as the one that approximates the exact dispersion relation for a range of wavenumbers while regarding other modes as spurious. Recently, Moura et al. [27] provided new interpretations on the role of spurious or secondary modes. In their work, these modes are replicates of the physical-mode along the wavenumber axis and they improve the accuracy of the scheme. Vanharen et al. [29] concluded that after a large number of iterations, high-order schemes behave in dispersion and dissipation according to the 35 physical-mode asymptotically, for wavenumbers less than π. Nevertheless, the complete behavior of DG-type high-order schemes in dispersion and dissipation based on all eigenmodes has not been studied before. In this paper, we provide a first attempt to achieve this goal.Whilst there exists abundant work on the analysis of both high-order and low-order schemes or classical finite difference/finite-volume schemes, little attention was given to comparing the DG, FD, and CD schemes 40 of the same order of accuracy. The DG method, originally introduced by Reed and Hill [30] to solve the neutron transport equation, is chosen in this study as a representative of the high-order polynomialbased methods capable of handling unstructured grids including the spectral difference (SD) [31], and the flux reconstruction (FR) or correction procedure via reconstruction (CPR) methods [32]. LaSaint and Raviart [33] performed an error analysis for the DG method. It was then further developed for convection-45 dominated problems and fluid dynamics by many researches, see for example ([34, 35, 36,...

show abstract

Section: True Behavior Of Dg Schemes Through a Combined-mode Semi-dismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Fourier analysis and evaluation of DG, FD and compact difference methods for conservation laws

Alhawwary

Wang

2018

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

“…In these regions, optimal meshes are generally pseudo-structured and highly aligned with the boundary. 3 Moreover, in the last decade high-order resolution methods (continuous Galerkin, 7,8 discontinuous Galerkin, 16 spectral differences, 30 k-exact, 14 ...) are more and more used. To preserve the high-order of convergence of these methods, it is required to have a high-order representation of the geometry in the mesh.…”

Section: Introductionmentioning

confidence: 99%

Connectivity-change moving mesh methods for high-order meshes: Toward closed advancing-layer high-order boundary layer mesh generation

Feuillet

Loseille

Marcum

et al. 2018

2018 Fluid Dynamics Conference

View full text Add to dashboard Cite

To cite this version:Rémi Feuillet, Adrien Loseille, David Marcum, Frédéric Alauzet. Connectivity-change moving mesh methods for high-order meshes: Toward closed advancing-layer high-order boundary layer mesh generation.Curved mesh generation starting from a P 1 mesh and closed advancing-layer boundary layer mesh generation both rely on mesh deformation and mesh optimization techniques. The approach presented in this work is to generalize connectivity-change moving mesh methods to high-order meshes. This approach is based on a high-order linear elasticity solver for the mesh deformation and on high-order mesh optimization operators such as mesh smoothing and generalized swapping. Thanks to this method, P k meshes are generated from P 1 meshes and closed advancing-layer boundary layer mesh generation will soon be possible.

show abstract

P2 Mesh Optimization Operators

Feuillet¹,

Loseille²,

Alauzet³

2019

Lecture Notes in Computational Science and Engineering

View full text Add to dashboard Cite

Curved mesh generation starting from a P 1 mesh relies on mesh deformation and mesh optimization techniques. Mesh optimization techniques consist in locally modifying the mesh in order to improve it with respect to a given quality criterion. This work presents the generalization of two mesh quality-based optimization operators to P 2 meshes. The generalized operators consist in mesh smoothing and generalized swapping. With the use of these operators, P 2 mesh generation starting from a P 1 mesh is more robust and P 2 connectivity-change moving mesh methods for large displacements are now possible.

show abstract

Revisiting the spectral analysis for high-order spectral discontinuous methods

Cited by 43 publications

References 34 publications

Fourier analysis and evaluation of DG, FD and compact difference methods for conservation laws

Fourier analysis and evaluation of DG, FD and compact difference methods for conservation laws

Connectivity-change moving mesh methods for high-order meshes: Toward closed advancing-layer high-order boundary layer mesh generation

P2 Mesh Optimization Operators

Contact Info

Product

Resources

About