{"title":"Uniform and optimal error estimates of a nested Picard integrator for the nonlinear Schrödinger equation with wave operator","authors":"Yongyong Cai, Yue Feng, Yichen Guo, Zhiguo Xu","doi":"10.1002/num.23135","DOIUrl":null,"url":null,"abstract":"We propose a second‐order nested Picard iterative integrator sine pseudospectral (NPI‐SP) method for the nonlinear Schrödinger equation with wave operator (NLSW) involving a parameter and carry out rigorous error estimates. Actually, the equation propagates wave with wavelength in time, while the amplitude of the leading oscillation is for well‐prepared initial data, and for ill‐prepared initial data, respectively. Based on the exponential integrator and nested Picard iteration, the uniformly accurate (w.r.t. ) NPI‐SP scheme is proposed with the optimal uniform error bounds at in time and spectral accuracy in space for both well‐prepared and ill‐prepared data in ‐norm. This result significantly improves the error bounds of the finite difference methods and exponential wave integrator for the NLSW. Error estimates are rigorously carried out and numerical examples are provided to confirm the theoretical analysis.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1000,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Numerical Methods for Partial Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1002/num.23135","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We propose a second‐order nested Picard iterative integrator sine pseudospectral (NPI‐SP) method for the nonlinear Schrödinger equation with wave operator (NLSW) involving a parameter and carry out rigorous error estimates. Actually, the equation propagates wave with wavelength in time, while the amplitude of the leading oscillation is for well‐prepared initial data, and for ill‐prepared initial data, respectively. Based on the exponential integrator and nested Picard iteration, the uniformly accurate (w.r.t. ) NPI‐SP scheme is proposed with the optimal uniform error bounds at in time and spectral accuracy in space for both well‐prepared and ill‐prepared data in ‐norm. This result significantly improves the error bounds of the finite difference methods and exponential wave integrator for the NLSW. Error estimates are rigorously carried out and numerical examples are provided to confirm the theoretical analysis.
期刊介绍：
An international journal that aims to cover research into the development and analysis of new methods for the numerical solution of partial differential equations, it is intended that it be readily readable by and directed to a broad spectrum of researchers into numerical methods for partial differential equations throughout science and engineering. The numerical methods and techniques themselves are emphasized rather than the specific applications. The Journal seeks to be interdisciplinary, while retaining the common thread of applied numerical analysis.
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