This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the n...This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the nonlinear term is mere cubic.The new framework of convergence analysis consists of two steps.In the first step,by truncating the nonlinear term into a global Lipschitz function,an alternative numerical method is proposed and proved in a rigorous way to be convergent in the discrete L2 norm;followed in the second step,the maximum bound of the numerical solution of the alternative numerical method is obtained by using a lifting technique,as implies that the two numerical methods are the same one.Under our framework of convergence analysis,with neither any restriction on the grid ratio nor any requirement of the small initial value,we establish the error estimate of the proposed conservative Fourier pseudo-spectral method,while previous work requires the certain restriction for the focusing case.The error bound is proved to be of O(h^(r)+t^(2))with grid size h and time step t.In fact,the framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the nonlinear Schr¨odinger-type equations.Numerical results are conducted to indicate the accuracy and efficiency of the proposed method,and investigate the effect of the nonlinear term and initial data on the blow-up solution.展开更多
为了系统分析产品社会生命周期评价的研究进展和技术框架,采用CiteSpace对来源于中国知网、Web of Science数据库的文献进行计量分析。结果表明:产品社会生命周期评价研究热点演进分为框架搭建(2008—2013年)、方法探索(2013—2016年)...为了系统分析产品社会生命周期评价的研究进展和技术框架,采用CiteSpace对来源于中国知网、Web of Science数据库的文献进行计量分析。结果表明:产品社会生命周期评价研究热点演进分为框架搭建(2008—2013年)、方法探索(2013—2016年)和体系标准化(2017—2020年)3个阶段;产品社会生命周期评价技术框架包括确定目标与范围、清单分析、影响评价与结果解释4个部分,清单分析的指标选择过程涉及40个子指标,其中与健康、人权相关的为核心指标;社会影响特征化可以通过参考量表法与影响路径法进行量化,参考量表法包括子类别评价法与社会热点数据库法,其中子类别评价法体系较为完善。展开更多
基金Jialing Wang’s work is supported by the National Natural Science Foundation of China(Grant No.11801277)Tingchun Wang’s work is supported by the National Natural Science Foundation of China(Grant No.11571181)+1 种基金the Natural Science Foundation of Jiangsu Province(Grant No.BK20171454)Qing Lan Project.Yushun Wang’s work is supported by the National Natural Science Foundation of China(Grant Nos.11771213 and 12171245).
文摘This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the nonlinear term is mere cubic.The new framework of convergence analysis consists of two steps.In the first step,by truncating the nonlinear term into a global Lipschitz function,an alternative numerical method is proposed and proved in a rigorous way to be convergent in the discrete L2 norm;followed in the second step,the maximum bound of the numerical solution of the alternative numerical method is obtained by using a lifting technique,as implies that the two numerical methods are the same one.Under our framework of convergence analysis,with neither any restriction on the grid ratio nor any requirement of the small initial value,we establish the error estimate of the proposed conservative Fourier pseudo-spectral method,while previous work requires the certain restriction for the focusing case.The error bound is proved to be of O(h^(r)+t^(2))with grid size h and time step t.In fact,the framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the nonlinear Schr¨odinger-type equations.Numerical results are conducted to indicate the accuracy and efficiency of the proposed method,and investigate the effect of the nonlinear term and initial data on the blow-up solution.
文摘为了系统分析产品社会生命周期评价的研究进展和技术框架,采用CiteSpace对来源于中国知网、Web of Science数据库的文献进行计量分析。结果表明:产品社会生命周期评价研究热点演进分为框架搭建(2008—2013年)、方法探索(2013—2016年)和体系标准化(2017—2020年)3个阶段;产品社会生命周期评价技术框架包括确定目标与范围、清单分析、影响评价与结果解释4个部分,清单分析的指标选择过程涉及40个子指标,其中与健康、人权相关的为核心指标;社会影响特征化可以通过参考量表法与影响路径法进行量化,参考量表法包括子类别评价法与社会热点数据库法,其中子类别评价法体系较为完善。
