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Dynamical behavior of a nondiffusive scheme for the advection equation
Nina Aguillon1 ; Pierre-Antoine Guihéneuf2
1 Sorbonne-Université, CNRS, Université de Paris, INRIA, Laboratoire Jacques-Louis Lions (LJLL), équipe ANGE, F-75005 Paris, France
2 Sorbonne Université, Université de Paris, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, F-75005 Paris, France
Confluentes Mathematici, Tome 12 (2020) no. 1, pp. 3-29.

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.

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DOI : 10.5802/cml.60
Classification : 37M10, 65M15, 65P40
Mots clés : 1-dimensional transport equation, nondiffusive scheme, dynamics, asymptotic behavior
Nina Aguillon 1 ; Pierre-Antoine Guihéneuf 2

1 Sorbonne-Université, CNRS, Université de Paris, INRIA, Laboratoire Jacques-Louis Lions (LJLL), équipe ANGE, F-75005 Paris, France
2 Sorbonne Université, Université de Paris, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, F-75005 Paris, France
Licence : CC-BY-NC-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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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/articles/10.5802/cml.60/

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