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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, Volume 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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/cml.60
Classification: 37M10, 65M15, 65P40
Keywords: 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
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Nina Aguillon; Pierre-Antoine Guihéneuf. Dynamical behavior of a nondiffusive scheme for the advection equation. Confluentes Mathematici, Volume 12 (2020) no. 1, pp. 3-29. doi : 10.5802/cml.60. https://cml.centre-mersenne.org/articles/10.5802/cml.60/

[1] Harald Bohr Almost Periodic Functions, Chelsea Publishing Company, New York, N.Y., 1947, ii+114 pages | MR

[2] Benjamin Boutin; Christophe Chalons; Frédéric Lagoutière; Philippe G. LeFloch Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I, Interfaces Free Bound., Volume 10 (2008) no. 3, pp. 399-421 | DOI | MR | Zbl

[3] Bruno Després Finite volume transport schemes, Numer. Math., Volume 108 (2008) no. 4, pp. 529-556 | DOI | MR | Zbl

[4] Bruno Després Stability of high order finite volume schemes for the 1D transport equation, Finite volumes for complex applications V, ISTE, London, 2008, pp. 337-342 | MR | Zbl

[5] Bruno Després; Frédéric Lagoutière Contact discontinuity capturing schemes for linear advection and compressible gas dynamics, J. Sci. Comput., Volume 16 (2001) no. 4, p. 479-524 (2002) | DOI | MR | Zbl

[6] Randall J. LeVeque Numerical methods for conservation laws, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992, x+214 pages | DOI | MR | Zbl

[7] John Milnor Correction and remarks: “On the concept of attractor”, Comm. Math. Phys., Volume 102 (1985) no. 3, pp. 517-519 http://projecteuclid.org/euclid.cmp/1104114467 | DOI | MR

[8] John Milnor On the concept of attractor, Comm. Math. Phys., Volume 99 (1985) no. 2, pp. 177-195 http://projecteuclid.org/euclid.cmp/1103942677 | DOI | MR | Zbl

[9] John Milnor Attractor, Scholarpedia, Volume 1 (2006) no. 11, p. 1815 (revision #186525) | DOI

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