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 · Wed Jun 17 00:00:00 EDT 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:: French National Research Agency; USDOE National Nuclear Security Administration (NNSA)

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

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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

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References (14)

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