Finite difference approximation for nonlinear Schrödinger equations with application to blow-up computation

Norikazu Saito, Takiko Sasaki

研究成果: Article査読

4 被引用数 (Scopus)

抄録

This paper presents a coherent analysis of the finite difference method to nonlinear Schrödinger (NLS) equations in one spatial dimension. We use the discrete H1 framework to establish well-posedness and error estimates in the L norm. The nonlinearity f(u) of a NLS equation is assumed to satisfy only a growth condition. We apply our results to computation of blow-up solutions for a NLS equation with the nonlinearity f(u) = - | u| 2p, p being a positive real number. Particularly, we offer the numerical blow-up time T(h, τ) , where h and τ are discretization parameters of space and time variables. We prove that T(h, τ) converges to the blow-up time T of the solution of the original NLS equation. Several numerical examples are presented to confirm the validity of theoretical results. Furthermore, we infer from numerical investigation that the convergence of T(h, τ) is at a second order rate in τ if the Crank–Nicolson scheme is applied to time discretization.

本文言語English
ページ(範囲)427-470
ページ数44
ジャーナルJapan Journal of Industrial and Applied Mathematics
33
2
DOI
出版ステータスPublished - 2016 7月 1
外部発表はい

ASJC Scopus subject areas

  • 工学(全般)
  • 応用数学

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