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Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations

Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations

Authors :
Enrique Zuazua
Liviu I. Ignat
Source :
BIRD: BCAM's Institutional Repository Data, instname
Publication Year :
2012

Abstract

This article is devoted to the analysis of the convergence rates of several numerical approximation schemes for linear and nonlinear Schrodinger equations on the real line. Recently, the authors have introduced viscous and two-grid numerical approximation schemes that mimic at the discrete level the so-called Strichartz dispersive estimates of the continuous Schrodinger equation. This allows to guarantee the convergence of numerical approximations for initial data in L 2 ( R ) , a fact that cannot be proved in the nonlinear setting for standard conservative schemes unless more regularity of the initial data is assumed. In the present article we obtain explicit convergence rates and prove that dispersive schemes fulfilling the Strichartz estimates are better behaved for H s ( R ) data if 0 s 1 / 2 . Indeed, while dispersive schemes ensure a polynomial convergence rate, non-dispersive ones only yield logarithmic ones.

Details

Database :
OpenAIRE
Journal :
BIRD: BCAM's Institutional Repository Data, instname
Accession number :
edsair.doi.dedup.....1e0fce3a2fa4f24c92db99ba7571c3d6