Comptes Rendus
no. 1-2
Mathematical Physics
High frequency wave packets for the Schrödinger equation and its numerical approximations
[Paquets d'ondes à haute fréquence pour l'équation de Schrödinger et ses approximations numériques]

Présenté par : Roland Glowinski

Aurora Marica1 ; Enrique Zuazua2, 1
1 BCAM - Basque Center for Applied Mathematics, Bizkaia Technology Park 500, 48160, Derio, Basque Country, Spain
2 Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011, Bilbao, Basque Country, Spain
Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 105-110.

On construit des paquets d'ondes gaussiennes pour l'équation de Schrödinger linéaire continue unidimensionnelle ainsi que pour sa semi-discrétisation en espace par différences finies. On illustre numériquement le manque d'uniformité par rapport au pas du maillage des propriétés de dispersion des solutions numériques démontré dans Ignat et Zuazua (2009) [6]. Par ailleurs, il est bien connu que les algorithmes bi-grilles sont des mécanismes de filtrage efficaces pour récupérer l'uniformité des propriétés dispersives. On analyse la façon dont les solutions bi-grilles correspondant à plusieurs projections de la grille fine sur la grossière se divisent en plusieurs paquets d'ondes, chacun se propageant différement. On représente numériquement ces phénomènes et on montre que ce comportement est en accord avec les résultats théoriques connus sur la dispersion des solutions bi-grilles.

We build Gaussian wave packets for the linear Schrödinger equation and its finite difference space semi-discretization and illustrate the lack of uniform dispersive properties of the numerical solutions as established in Ignat and Zuazua (2009) [6]. It is by now well known that bigrid algorithms provide filtering mechanisms allowing to recover the uniformity of the dispersive properties as the mesh size goes to zero. We analyze and illustrate numerically how these high frequency wave packets split and propagate under these bigrid filtering mechanisms, depending on how the fine grid/coarse grid filtering is implemented.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.11.009
Aurora Marica 1 ; Enrique Zuazua 2, 1

1 BCAM - Basque Center for Applied Mathematics, Bizkaia Technology Park 500, 48160, Derio, Basque Country, Spain
2 Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011, Bilbao, Basque Country, Spain 
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     author = {Aurora Marica and Enrique Zuazua},
     title = {High frequency wave packets for the {Schr\"odinger} equation and its numerical approximations},
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     language = {en},
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Aurora Marica; Enrique Zuazua. High frequency wave packets for the Schrödinger equation and its numerical approximations. Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 105-110. doi : 10.1016/j.crma.2010.11.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.11.009/

[1] A. Arnold; M. Ehrhardt; I. Sofronov Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation and stability, Comm. Math. Sci., Volume 1 (2003) no. 3, pp. 501-556

[2] T. Cazenave Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, AMS, New York, 2003

[3] E. Faou, B. Grébert, Hamiltonian interpolation of splitting approximations for nonlinear PDEs, preprint.

[4] R. Glowinski Ensuring the well-posedness by analogy; Stokes problem and boundary control for the wave equation, Journal of Computational Physics, Volume 103 (1992) no. 2, pp. 189-221

[5] L. Ignat, A splitting method for the non-linear Schrödinger equation, preprint.

[6] L. Ignat; E. Zuazua Numerical dispersive schemes for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., Volume 47 (2009) no. 2, pp. 1366-1390

[7] L. Ignat, E. Zuazua, Convergence rates for dispersive approximation schemes for non-linear Schrödinger equations, preprint.

[8] C.K. Linares; G. Ponce Introduction to Nonlinear Dispersive Equations, Springer, 2009

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