Convergence rates for dispersive approximation schemes to nonlinear Schr"odinger equations
Convergence rates for dispersive approximation schemes to nonlinear Schr"odinger equations
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非线性Schr"odinger方程的色散近似格式的收敛率
DOI：
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发表时间：
2011
期刊：
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通讯作者：
Enrique Zuazua
中科院分区：
文献类型：
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作者：
L. Ignat;Enrique Zuazua
本文分析了线性和非线性Schr“odinger方程在真实的直线上的几种数值逼近格式的收敛速度。最近，作者介绍了粘性和两个网格的数值逼近方案，在离散水平上模仿所谓的连续薛定谔方程的Schr“odinger色散估计。这允许保证收敛的初始数据在L2（R），一个事实，不能证明在非线性设置的标准保守计划，除非更多的规则性的初始数据被假定。本文得到了显式的收敛速度，并证明了当0实际上，虽然分散方案确保多项式收敛率，但非分散方案仅产生对数衰减率。
This article is devoted to the analysis of the convergence rates of several nu- merical approximation schemes for linear and nonlinear Schr"odinger 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 Schr"odinger equation. This allows to guarantee the convergence of numerical approximations for initial data in L2(R), a fact that can not 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 Hs(R) data if 0<s<1/2. Indeed, while dispersive schemes ensure a polynomial convergence rate, non-dispersive ones only yield logarithmic decay rates.