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Le Roux, Daniel Y.

Fourier analysis of the local discontinuous Galerkin method for the linearized KdV equation. (English) Zbl 1502.35145

GEM. Int. J. Geomath. 13, Paper No. 19, 27 p. (2022).
Summary: 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.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
74S05 Finite element methods applied to problems in solid mechanics
35L99 Hyperbolic equations and hyperbolic systems
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs

Keywords:

Korteweg-de Vries equation; dispersive waves; discontinuous Galerkin methods; finite-element method; Fourier analysis; eigenvalue problems
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References:

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