Dynamical behavior of a nondiffusive scheme for the advection equation
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.

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/

[1] Harald Bohr Almost Periodic Functions, Chelsea Publishing Company, New York, N.Y., 1947, ii+114 pages | MR 0020163 (8,512a)

[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 | Article | MR 2453138 | Zbl 1157.65435

[3] Bruno Després Finite volume transport schemes, Numer. Math., Volume 108 (2008) no. 4, pp. 529-556 | Article | MR 2369203 | Zbl 1140.65058

[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 2451425 | Zbl 1422.65204

[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) | Article | MR 1881855 | Zbl 0999.76091

[6] Randall J. LeVeque Numerical methods for conservation laws, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992, x+214 pages | Article | MR 1153252 | Zbl 0847.65053

[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 | Article | MR 818833

[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 | Article | MR 790735 | Zbl 0595.58028

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