Numerical Analysis/Partial Differential Equations
A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence
Comptes Rendus. Mathématique, Volume 341 (2005) no. 6, pp. 381-386.

We introduce a two-grid finite difference approximation scheme for the free Schrödinger equation. This scheme is shown to converge and to posses appropriate dispersive properties as the mesh-size tends to zero. A careful analysis of the Fourier symbol shows that this occurs because the two-grid algorithm (consisting in projecting slowly oscillating data into a fine grid) acts, to some extent, as a filtering one. We show that this scheme converges also in a class of nonlinear Schrödinger equations whose well-posedness analysis requires the so-called Strichartz estimates. This method provides an alternative to the method introduced by the authors [L.I. Ignat, E. Zuazua, Dispersive properties of a viscous numerical scheme for the Schrödinger equation, C. R. Math. Acad. Sci. Paris 340 (7) (2005) 529–534] using numerical viscosity.

On introduit une méthode bi-maille semi-discrète en différences finies pour l'approximation numérique de l'équation de Schrödinger. On démontre la convergence L2 du schéma et des propriétés dispersives uniformes par rapport au pas du maillage. Une analyse soigneuse en Fourier du symbole du schéma (consistant essentiellement à projeter des données lentes sur un maillage fin) montre que l'algorithme bi-maille agit comme un filtre des hautes fréquences. On montre aussi la convergence du schéma dans une classe d'équations non-linéaires dont l'étude dans le cas continu nécessite des inégalités de Strichartz. Cette méthode donne une approche alternative à celle introduite par les auteurs [L.I. Ignat, E. Zuazua, Dispersive properties of a viscous numerical scheme for the Schrödinger equation, C. R. Math. Acad. Sci. Paris 340 (7) (2005) 529–534] à l'aide d'un schéma avec viscosité numérique.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.07.018
Ignat, Liviu I. 1; Zuazua, Enrique 1

1 Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain
@article{CRMATH_2005__341_6_381_0,
     author = {Ignat, Liviu I. and Zuazua, Enrique},
     title = {A two-grid approximation scheme for nonlinear {Schr\"odinger} equations: dispersive properties and convergence},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {381--386},
     publisher = {Elsevier},
     volume = {341},
     number = {6},
     year = {2005},
     doi = {10.1016/j.crma.2005.07.018},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2005.07.018/}
}
TY  - JOUR
AU  - Ignat, Liviu I.
AU  - Zuazua, Enrique
TI  - A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 381
EP  - 386
VL  - 341
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2005.07.018/
DO  - 10.1016/j.crma.2005.07.018
LA  - en
ID  - CRMATH_2005__341_6_381_0
ER  - 
%0 Journal Article
%A Ignat, Liviu I.
%A Zuazua, Enrique
%T A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence
%J Comptes Rendus. Mathématique
%D 2005
%P 381-386
%V 341
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2005.07.018/
%R 10.1016/j.crma.2005.07.018
%G en
%F CRMATH_2005__341_6_381_0
Ignat, Liviu I.; Zuazua, Enrique. A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence. Comptes Rendus. Mathématique, Volume 341 (2005) no. 6, pp. 381-386. doi : 10.1016/j.crma.2005.07.018. http://www.numdam.org/articles/10.1016/j.crma.2005.07.018/

[1] Cazenave, T. Semilinear Schrödinger Equations, Courant Lecture Notes, vol. 10, Amer. Math. Soc., Providence, RI, 2003

[2] Glowinski, R. Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation, J. Comput. Phys., Volume 103 (1992) no. 2, pp. 189-221

[3] Ignat, L.I.; Zuazua, E. Dispersive properties of a viscous numerical scheme for the Schrödinger equation, C. R. Math. Acad. Sci. Paris, Volume 340 (2005) no. 7, pp. 529-534

[4] Keel, M.; Tao, T. Endpoint Strichartz estimates, Amer. J. Math., Volume 120 (1998) no. 5, pp. 955-980

[5] Kenig, C.E.; Ponce, G.; Vega, L. Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., Volume 40 (1991) no. 1, pp. 33-69

[6] Negreanu, M.; Zuazua, E. Convergence of a multigrid method for the controllability of a 1-d wave equation, C. R. Math. Acad. Sci. Paris, Ser. I, Volume 338 (2004) no. 5, pp. 413-418

[7] Stefanov, A.; Kevrekidis, P.G. Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein–Gordon equations, Nonlinearity, Volume 18 (2005), pp. 1841-1857

[8] Tsutsumi, Y. L2-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcional Ekvacioj Ser. Int., Volume 30 (1987), pp. 115-125

Cited by Sources: