Fourier analysis of the local discontinuous Galerkin method for the linearized KdV equation

A Fourier/stability analysis of the third-order Korteweg–de Vries equation is presented subject to a class of local discontinuous Galerkin discretization using high-degree Lagrange polynomials. The selection of stability parameters involved in the method is made on the basis of the study of the high...

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Published inGEM international journal on geomathematics Vol. 13; no. 1; pp. 1845 - 1866
Main Author Le Roux, Daniel Y.
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2022
Springer
Subjects
Applications of Mathematics
Computational Science and Engineering
Earth Sciences
Mathematics
Mathematics and Statistics
Numerical Methods for Global and Regional Ocean Modeling
Original Paper
Discontinuous Galerkin methods
74S05
Eigenvalue problems
Korteweg–de Vries equation
76B15
Finite-element method
Fourier analysis
35Q53
Dispersive waves
35P20
35L99
65M12
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Abstract A Fourier/stability analysis of the third-order Korteweg–de Vries equation is presented subject to a class of local discontinuous Galerkin discretization using high-degree Lagrange polynomials. The selection of stability parameters involved in the method is made on the basis of the study of the higher frequency eigenmodes and the Fourier analysis. Explicit analytical dispersion relation and group velocity are obtained and the stability study of the discrete frequency is performed. The emergence of gaps in the imaginary part of the computed frequency is observed and studied for the first time to our knowledge. Further, a superconvergent result is demonstrated for the discrete frequency by obtaining an explicit analytical asymptotic formula for the latter.
AbstractList A Fourier/stability analysis of the third-order Korteweg–de Vries equation is presented subject to a class of local discontinuous Galerkin discretization using high-degree Lagrange polynomials. The selection of stability parameters involved in the method is made on the basis of the study of the higher frequency eigenmodes and the Fourier analysis. Explicit analytical dispersion relation and group velocity are obtained and the stability study of the discrete frequency is performed. The emergence of gaps in the imaginary part of the computed frequency is observed and studied for the first time to our knowledge. Further, a superconvergent result is demonstrated for the discrete frequency by obtaining an explicit analytical asymptotic formula for the latter.
ArticleNumber 19
Author Le Roux, Daniel Y.
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  organization: CNRS, Université Lyon 1, Institut Camille Jordan, Université de Lyon
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10.1137/S0036142901390378
10.1016/j.jcp.2005.10.016
10.1006/jcph.1999.6227
10.1016/j.cam.2013.06.004
10.1007/s10915-017-0377-z
10.1007/978-0-387-72067-8
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Issue 1
Keywords Discontinuous Galerkin methods
74S05
Eigenvalue problems
Korteweg–de Vries equation
76B15
Finite-element method
Fourier analysis
35Q53
Dispersive waves
35P20
35L99
65M12
Language English
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SubjectTerms Applications of Mathematics
Computational Science and Engineering
Earth Sciences
Mathematics
Mathematics and Statistics
Numerical Methods for Global and Regional Ocean Modeling
Original Paper
Title Fourier analysis of the local discontinuous Galerkin method for the linearized KdV equation
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