Ghost solutions with centered schemes for one-dimensional transport equations with Neumann boundary conditions
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 29 (2020) no. 4, pp. 927-950.

This paper is devoted to the space-centered θ-scheme for the transport equation with constant velocity on an interval, with homogeneous Neumann boundary data on each side. The numerical solution presents a weird behavior, as if the initial datum was periodical over , with a period that would be twice or four times (depending on the case) the length of the considered interval. We precisely formulate this statement and prove it. We also study a similar behavior in the case of Dirichlet boundary conditions.

Dans cet article, nous nous intéressons au θ-schéma centré en espace pour la résolution approchée de l’équation de transport à vitesse constante dans un segment en dimension 1, avec données de Neumann homogènes aux deux bords du domaine. De manière très surprenante, la solution numérique présente un caractère périodique, comme si la donnée initiale était périodique sur avec une période égale à deux fois ou quatre fois, suivant les cas, la longueur du segment. Nous énonçons précisément ce résultat et le démontrons, puis évoquons un comportement similaire dans le cas où les deux conditions de bord sont de type Dirichlet.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1650

Mélanie Inglard 1; Frédéric Lagoutière 2; Hans Henrik Rugh 3

1 Lycée Saint-Louis, 44 boulevard Saint-Michel, 75006 Paris (France)
2 Université de Lyon, Université Claude Bernard Lyon 1, Institut Camille Jordan (CNRS UMR5208), 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex (France)
3 Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay (France)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AFST_2020_6_29_4_927_0,
     author = {M\'elanie Inglard and Fr\'ed\'eric Lagouti\`ere and Hans Henrik Rugh},
     title = {Ghost solutions with centered schemes for one-dimensional transport equations with {Neumann} boundary conditions},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {927--950},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 29},
     number = {4},
     year = {2020},
     doi = {10.5802/afst.1650},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1650/}
}
TY  - JOUR
AU  - Mélanie Inglard
AU  - Frédéric Lagoutière
AU  - Hans Henrik Rugh
TI  - Ghost solutions with centered schemes for one-dimensional transport equations with Neumann boundary conditions
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2020
SP  - 927
EP  - 950
VL  - 29
IS  - 4
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1650/
DO  - 10.5802/afst.1650
LA  - en
ID  - AFST_2020_6_29_4_927_0
ER  - 
%0 Journal Article
%A Mélanie Inglard
%A Frédéric Lagoutière
%A Hans Henrik Rugh
%T Ghost solutions with centered schemes for one-dimensional transport equations with Neumann boundary conditions
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2020
%P 927-950
%V 29
%N 4
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1650/
%R 10.5802/afst.1650
%G en
%F AFST_2020_6_29_4_927_0
Mélanie Inglard; Frédéric Lagoutière; Hans Henrik Rugh. Ghost solutions with centered schemes for one-dimensional transport equations with Neumann boundary conditions. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 29 (2020) no. 4, pp. 927-950. doi : 10.5802/afst.1650. https://afst.centre-mersenne.org/articles/10.5802/afst.1650/

[1] Xavier Antoine; Anton Arnold; Christophe Besse; Matthias Ehrhardt; Achim Schädle A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comput. Phys., Volume 4 (2008) no. 4, pp. 729-796 | Zbl

[2] Benjamin Boutin; Jean François Coulombel Stability of finite difference schemes for hyperbolic initial boundary value problems: Numerical boundary layers, Numer. Math., Theory Methods Appl., Volume 10 (2017) no. 3, pp. 489-519 | DOI | MR | Zbl

[3] Jean-François Coulombel; Frédéric Lagoutière The Neumann numerical boundary condition for transport equations (2018) (https://hal.archives-ouvertes.fr/hal-01902551) | Zbl

[4] Matthias Ehrhardt Absorbing boundary conditions for hyperbolic systems, Numer. Math., Theory Methods Appl., Volume 3 (2010) no. 3, pp. 295-337 | DOI | MR | Zbl

[5] Bertil Gustafsson The convergence rate for difference approximations to mixed initial boundary value problems, Math. Comput., Volume 29 (1975), pp. 396-406 | DOI | MR | Zbl

[6] Bertil Gustafsson The convergence rate for difference approximations to general mixed initial-boundary value problems, SIAM J. Numer. Anal., Volume 18 (1981) no. 2, pp. 179-190 | DOI | MR | Zbl

[7] Bertil Gustafsson; Heinz-Otto Kreiss; Joseph Oliger Time dependent problems and difference methods, Pure and Applied Mathematics, John Wiley & Sons, 1995, xii+642 pages | Zbl

[8] Bertil Gustafsson; Heinz-Otto Kreiss; Arne Sundström Stability theory of difference approximations for mixed initial boundary value problems. II, Math. Comput., Volume 26 (1972), pp. 649-686 | DOI | MR | Zbl

[9] Heinz-Otto Kreiss; Einar Lundqvist On difference approximations with wrong boundary values, Math. Comput., Volume 22 (1968), pp. 1-12 | DOI | MR | Zbl

[10] Randall J. LeVeque Finite difference methods for ordinary and partial differential equations. Steady-state and time-dependent problems, Society for Industrial and Applied Mathematics, 2007, xvi+341 pages | Zbl

[11] Lloyd N. Trefethen Instability of difference models for hyperbolic initial-boundary value problems, Commun. Pure Appl. Math., Volume 37 (1984) no. 3, pp. 329-367 | DOI | MR | Zbl

Cited by Sources: