Fourier Analyses of High-Order Continuous and Discontinuous Galerkin Methods

Le Roux, Daniel Y.; Eldred, Christopher; Taylor, Mark A.

doi:10.1137/19m1289595

Fourier Analyses of High-Order Continuous and Discontinuous Galerkin Methods

Journal Article · 17 June 2020 · SIAM Journal on Numerical Analysis

DOI:https://doi.org/10.1137/19m1289595· OSTI ID:1725844

Le Roux, Daniel Y. ^[1]; Eldred, Christopher ^[2]; Taylor, Mark A. ^[3]

Univ. of Lyon (France)
Univ. of Grenoble (France)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

In this paper, we present a Fourier analysis of wave propagation problems subject to a class of continuous and discontinuous discretizations using high-degree Lagrange polynomials. This allows us to obtain explicit analytical formulas for the dispersion relation and group velocity and, for the first time to our knowledge, characterize analytically the emergence of gaps in the dispersion relation at specific wavenumbers, when they exist, and compute their specific locations. Wave packets with energy at these wavenumbers will fail to propagate correctly, leading to significant numerical dispersion. We also show that the Fourier analysis generates mathematical artifacts, and we explain how to remove them through a branch selection procedure conducted by analysis of eigenvectors and associated reconstructed solutions. The higher frequency eigenmodes, named erratic in this study, are also investigated analytically and numerically.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA); French National Research Agency

Grant/Contract Number:: AC04-94AL85000; NA0003525

OSTI ID:: 1725844

Report Number(s):: SAND--2020-11694J; 692226

Journal Information:: SIAM Journal on Numerical Analysis, Journal Name: SIAM Journal on Numerical Analysis Journal Issue: 3 Vol. 58; ISSN 0036-1429

Publisher:: Society for Industrial and Applied MathematicsCopyright Statement

Country of Publication:: United States

Language:: English

References (14)

Dispersion analysis of the spectral element method Melvin, Thomas; Staniforth, Andrew; Thuburn, John Quarterly Journal of the Royal Meteorological Society, Vol. 138, Issue 668 https://doi.org/10.1002/qj.1906	journal	February 2012
An Analysis of the Discontinuous Galerkin Method for Wave Propagation Problems Hu, Fang Q.; Hussaini, M. Y.; Rasetarinera, Patrick Journal of Computational Physics, Vol. 151, Issue 2 https://doi.org/10.1006/jcph.1999.6227	journal	May 1999
Eigensolution Analysis of the Discontinuous Galerkin Method with Nonuniform Grids Hu, Fang Q.; Atkins, Harold L. Journal of Computational Physics, Vol. 182, Issue 2 https://doi.org/10.1006/jcph.2002.7184	journal	November 2002
An analysis of the spectrum of the discontinuous Galerkin method Krivodonova, Lilia; Qin, Ruibin Applied Numerical Mathematics, Vol. 64 https://doi.org/10.1016/j.apnum.2012.07.008	journal	February 2013
Spatial eigensolution analysis of discontinuous Galerkin schemes with practical insights for under-resolved computations and implicit LES Mengaldo, G.; Moura, R. C.; Giralda, B. Computers & Fluids, Vol. 169 https://doi.org/10.1016/j.compfluid.2017.09.016	journal	June 2018
Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods Ainsworth, Mark Journal of Computational Physics, Vol. 198, Issue 1 https://doi.org/10.1016/j.jcp.2004.01.004	journal	July 2004
Dispersive behaviour of high order finite element schemes for the one-way wave equation Ainsworth, Mark Journal of Computational Physics, Vol. 259 https://doi.org/10.1016/j.jcp.2013.11.003	journal	February 2014
Linear dispersion–diffusion analysis and its application to under-resolved turbulence simulations using discontinuous Galerkin spectral/hp methods Moura, R. C.; Sherwin, S. J.; Peiró, J. Journal of Computational Physics, Vol. 298 https://doi.org/10.1016/j.jcp.2015.06.020	journal	October 2015
Dispersion analysis of compatible Galerkin schemes for the 1D shallow water model Eldred, Christopher; Le Roux, Daniel Y. Journal of Computational Physics, Vol. 371 https://doi.org/10.1016/j.jcp.2018.06.007	journal	October 2018
Impact and importance of hyperdiffusion on the spectral element method: A linear dispersion analysis Ullrich, Paul A.; Reynolds, Daniel R.; Guerra, Jorge E. Journal of Computational Physics, Vol. 375 https://doi.org/10.1016/j.jcp.2018.06.035	journal	December 2018
Dispersion analysis of compatible Galerkin schemes on quadrilaterals for shallow water models Eldred, Christopher; Le Roux, Daniel Y. Journal of Computational Physics, Vol. 387 https://doi.org/10.1016/j.jcp.2019.02.009	journal	June 2019
Spatial eigenanalysis of spectral/hp continuous Galerkin schemes and their stabilisation via DG-mimicking spectral vanishing viscosity for high Reynolds number flows Moura, Rodrigo C.; Aman, Mansoor; Peiró, Joaquim Journal of Computational Physics, Vol. 406 https://doi.org/10.1016/j.jcp.2019.109112	journal	April 2020
Determinants of Sums Marcus, Marvin The College Mathematics Journal, Vol. 21, Issue 2 https://doi.org/10.1080/07468342.1990.11973297	journal	March 1990
Superconvergence in Finite Element Methods and Meshes That are Locally Symmetric with Respect to a Point Schatz, A. H.; Sloan, I. H.; Wahlbin, L. B. SIAM Journal on Numerical Analysis, Vol. 33, Issue 2 https://doi.org/10.1137/0733027	journal	April 1996

Similar Records

Explicit dispersion relations for warm fluid waves in a uniform plasma (invited)

Journal Article · 2023 · Journal of the Korean Physical Society · OSTI ID:1991363

Optimal numerical parameterization of discontinuous Galerkin method applied to wave propagation problems

Journal Article · 2007 · Journal of Computational Physics · OSTI ID:20991571

Resonant effect of the noncircular shape of the plasma surface on the dispersion properties of extraordinary azimuthal surface modes in magnetoactive waveguides

Journal Article · 2007 · Plasma Physics Reports · OSTI ID:21100161

Related Subjects

97 MATHEMATICS AND COMPUTING
computational modes
discontinuous Galerkin method
dispersion analysis
erratic numerical modes
finite-element method
numerical dispersion

Fourier Analyses of High-Order Continuous and Discontinuous Galerkin Methods

Citation Formats

References (14)

Similar Records

Related Subjects