[La fonction de Green des schémas de Lax–Wendroff et Beam–Warming]
On obtient une borne Gaussienne généralisée pour la fonction de Green des schémas de Lax–Wendroff et Beam–Warming. Cette borne permet de préciser la région de l’espace qui conduit à l’instabilité bien connue de ces schémas pour la norme uniforme. On retrouve par ailleurs des bornes uniformes quand ces schémas sont appliqués à des suites à variations bornées.
We prove a sharp uniform generalized Gaussian bound for the Green’s function of the Lax–Wendroff and Beam–Warming schemes. Our bound highlights the spatial region that leads to the well-known (rather weak) instability of these schemes in the maximum norm. We also recover uniform bounds in the maximum norm when these schemes are applied to initial data of bounded variation.
Keywords: Transport equation, Lax–Wendroff scheme, Beam–Warming scheme, difference approximation, convolution, stability, local limit theorem
Mot clés : Équation de transport, schéma de Lax–Wendroff, schéma de Beam–Warming, approximation par différences finies, convolution, stabilité, théorème de la limite locale
Jean-François Coulombel 1
CC-BY 4.0
@article{AMBP_2022__29_2_247_0,
author = {Jean-Fran\c{c}ois Coulombel},
title = {The {Green{\textquoteright}s} function of the {Lax{\textendash}Wendroff} and {Beam{\textendash}Warming} schemes},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {247--294},
publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
volume = {29},
number = {2},
year = {2022},
doi = {10.5802/ambp.413},
language = {en},
url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.413/}
}
TY - JOUR AU - Jean-François Coulombel TI - The Green’s function of the Lax–Wendroff and Beam–Warming schemes JO - Annales mathématiques Blaise Pascal PY - 2022 SP - 247 EP - 294 VL - 29 IS - 2 PB - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal UR - https://ambp.centre-mersenne.org/articles/10.5802/ambp.413/ DO - 10.5802/ambp.413 LA - en ID - AMBP_2022__29_2_247_0 ER -
%0 Journal Article %A Jean-François Coulombel %T The Green’s function of the Lax–Wendroff and Beam–Warming schemes %J Annales mathématiques Blaise Pascal %D 2022 %P 247-294 %V 29 %N 2 %I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal %U https://ambp.centre-mersenne.org/articles/10.5802/ambp.413/ %R 10.5802/ambp.413 %G en %F AMBP_2022__29_2_247_0
Jean-François Coulombel. The Green’s function of the Lax–Wendroff and Beam–Warming schemes. Annales mathématiques Blaise Pascal, Tome 29 (2022) no. 2, pp. 247-294. doi : 10.5802/ambp.413. https://ambp.centre-mersenne.org/articles/10.5802/ambp.413/
[1] Asymptotic methods in analysis, Dover Publications, 1981
[2] Finite volume transport schemes, Numer. Math., Volume 108 (2008) no. 4, pp. 529-556
[3] Convolution powers of complex functions on , Math. Nachr., Volume 287 (2014) no. 10, pp. 1106-1130
[4] Boundedness of dispersive difference schemes, Math. Comput., Volume 55 (1990) no. 191, pp. 55-87
[5] Time dependent problems and difference methods, Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs & Tracts, John Wiley & Sons, 1995
[6] The near-stability of the Lax–Wendroff method, Numer. Math., Volume 7 (1965), pp. 73-77
[7] Norms of powers of absolutely convergent Fourier series, Mich. Math. J., Volume 13 (1966), pp. 393-416
[8] Séries de Fourier absolument convergentes, Ergebnisse der Mathematik und ihrer Grenzgebiete, 50, Springer, 1970
[9] A simple proof of Wiener’s theorem, Proc. Am. Math. Soc., Volume 48 (1975), pp. 264-265
[10] Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, 93, American Mathematical Society, 2002
[11] Sums of independent random variables, Ergebnisse der Mathematik und ihrer Grenzgebiete, 82, Springer, 1975
[12] On the convolution powers of complex functions on , J. Fourier Anal. Appl., Volume 21 (2015) no. 4, pp. 754-798 | Zbl
[13] Convolution powers of complex functions on , Rev. Mat. Iberoam., Volume 33 (2017) no. 3, pp. 1045-1121
[14] Difference methods for initial value problems. Theory and applications, Graduate Texts in Mathematics, Interscience Publishers, 1967 | Zbl
[15] Real and complex analysis, McGraw-Hill, 1987
[16] Existence and stability of traveling discrete shocks, Numerical analysis and its applications (Rousse, 1996) (Lecture Notes in Computer Science), Volume 1196, Springer, 1997, pp. 466-473
[17] A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions, Linear operators (Warsaw, 1994) (Banach Center Publications), Volume 38, Polish Academy of Sciences, 1997, pp. 339-360
[18] Stability of difference schemes in the maximum-norm, J. Differ. Equations, Volume 1 (1965), pp. 273-292
[19] Spectra and pseudospectra. The behavior of nonnormal matrices and operators, Princeton University Press, 2005 | Zbl
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