Dynamical behavior of a nondiffusive scheme for the advection equation

Nina Aguillon; Pierre-Antoine Guihéneuf

doi:10.5802/cml.60

Nina Aguillon ; Pierre-Antoine Guihéneuf

Confluentes Mathematici, Tome 12 (2020) no. 1, pp. 3-29.

Résumé

We study the long time behaviour of a dynamical system strongly linked to the nondiffusive scheme of Després and Lagoutiere for the $1$ -dimensional transport equation. This scheme is nondiffusive in the sense that discontinuities are not smoothened out through time. Numerical simulations indicate that the scheme error is uniformly bounded with time. We prove that this scheme is overcompressive when the Courant–Friedrichs–Levy number is $1 / 2$ : when the initial data is nondecreasing, the approximate solution becomes a Heaviside function. In a special case, we also understand how plateaus are formed in the solution and their stability, a distinctive feature of the Després and Lagoutière scheme.

Reçu le : 2019-09-10
Révisé le : 2020-04-14
Accepté le : 2020-06-23
Publié le : 2020-09-25

DOI : https://doi.org/10.5802/cml.60
Classification : 37M10, 65M15, 65P40
Mots clés: 1-dimensional transport equation, nondiffusive scheme, dynamics, asymptotic behavior

@article{CML_2020__12_1_3_0,
     author = {Nina Aguillon and Pierre-Antoine Guih\'eneuf},
     title = {Dynamical behavior of a nondiffusive scheme for the advection equation},
     journal = {Confluentes Mathematici},
     pages = {3--29},
     publisher = {Institut Camille Jordan},
     volume = {12},
     number = {1},
     year = {2020},
     doi = {10.5802/cml.60},
     language = {en},
     url = {cml.centre-mersenne.org/item/CML_2020__12_1_3_0/}
}

Nina Aguillon; Pierre-Antoine Guihéneuf. Dynamical behavior of a nondiffusive scheme for the advection equation. Confluentes Mathematici, Tome 12 (2020) no. 1, pp. 3-29. doi : 10.5802/cml.60. https://cml.centre-mersenne.org/item/CML_2020__12_1_3_0/

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