Ignat, Liviu I.; Zuazua, Enrique A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence. (English. Abridged French version) Zbl 1079.65090 C. R., Math., Acad. Sci. Paris 341, No. 6, 381-386 (2005). Summary: 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 [ibid. 340 (7), 529–534 (2005; Zbl 1036.35016)] using numerical viscosity. Cited in 16 Documents MSC: 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35Q55 NLS equations (nonlinear Schrödinger equations) Keywords:convergence; finite difference; Schrödinger equation; two-grid algorithm Citations:Zbl 1036.35016 PDFBibTeX XMLCite \textit{L. I. Ignat} and \textit{E. Zuazua}, C. R., Math., Acad. Sci. Paris 341, No. 6, 381--386 (2005; Zbl 1079.65090) Full Text: DOI References: [1] Cazenave, T., Semilinear Schrödinger Equations, Courant Lecture Notes, vol. 10 (2003), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1055.35003 [2] Glowinski, R., Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation, J. Comput. Phys., 103, 2, 189-221 (1992) · Zbl 0763.76042 [3] Ignat, L. I.; Zuazua, E., Dispersive properties of a viscous numerical scheme for the Schrödinger equation, C. R. Math. Acad. Sci. Paris, 340, 7, 529-534 (2005) · Zbl 1063.35016 [4] Keel, M.; Tao, T., Endpoint Strichartz estimates, Amer. J. Math., 120, 5, 955-980 (1998) · Zbl 0922.35028 [5] Kenig, C. E.; Ponce, G.; Vega, L., Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40, 1, 33-69 (1991) · Zbl 0738.35022 [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, 338, 5, 413-418 (2004) · Zbl 1038.65054 [7] Stefanov, A.; Kevrekidis, P. G., Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein-Gordon equations, Nonlinearity, 18, 1841-1857 (2005) · Zbl 1181.35266 [8] Tsutsumi, Y., \(L^2\)-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcional Ekvacioj Ser. Int., 30, 115-125 (1987) · Zbl 0638.35021 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.