An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods （TSCRK） is presented. Their construction, their order conditio...An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods （TSCRK） is presented. Their construction, their order conditions and their convergence are studied. The two-step continuity Runge-Kutta methods possess good numerical stability properties and higher stage-order, and keep the explicit process of computing the Runge-Kutta stages. The numerical experiments show that the TSCRK methods are efficient.展开更多
In this paper,a stochastic linear two-step scheme has been presented to approximate backward stochastic differential equations(BSDEs).A necessary and sufficient condition is given to judge the L 2-stability of our num...In this paper,a stochastic linear two-step scheme has been presented to approximate backward stochastic differential equations(BSDEs).A necessary and sufficient condition is given to judge the L 2-stability of our numerical schemes.This stochastic linear two-step method possesses a family of 3-order convergence schemes in the sense of strong stability.The coefficients in the numerical methods are inferred based on the constraints of strong stability and n-order accuracy(n∈N^(+)).Numerical experiments illustrate that the scheme is an efficient probabilistic numerical method.展开更多
The digital economy has become an important driver for stimulating economic growth.The digital economy has now widely penetrated the fields of economy and society,providing new opportunities for the develop‐ment of u...The digital economy has become an important driver for stimulating economic growth.The digital economy has now widely penetrated the fields of economy and society,providing new opportunities for the develop‐ment of urban-rural integration.Based on panel data for 30 provinces in China from 2011 to 2020,this study constructed an index system for the integration of the digital economy and the development of urban-rural areas and conducted a systematic measurement analysis.Additionally,we used a two-step system of GMM estimation to analyze the impact of the digital economy on the development of urban-rural integra‐tion.The findings demonstrate the significant imbalance paradox of China’s digital economy development,which is shown in a gradient where the eastern region is higher than the center and the central region is higher than the west.Urban-rural integration levels in China fluctuate and display geographical variance,typically displaying high levels in the east and low levels in the west.Urban-rural integration is significantly encouraged by the digital economy,yet it varies in variability between different areas and dimensions.Addi‐tionally,rural human capital moderates the favorable effects of the digital economy on urban-rural integra‐tion.As a result,in order to achieve the integrated development of urban and rural areas,it is imperative to fully exploit the active role of the digital economy,better support the development of rural revitalization,bridge the“digital divide”between urban and rural development,and build a strong foundation for the for‐mation of a digital urban-rural integrated development pattern with urban and rural areas and common con‐struction and sharing.展开更多
In this paper, a class of two-step continuity Runge-Kutta（TSCRK） methods for solving singular delay differential equations（DDEs） is presented. Analysis of numerical stability of this methods is given. We consider ...In this paper, a class of two-step continuity Runge-Kutta（TSCRK） methods for solving singular delay differential equations（DDEs） is presented. Analysis of numerical stability of this methods is given. We consider the two distinct cases： （i）τ≥ h, （ii）τ 〈 h, where the delay τ and step size h of the two-step continuity Runge-Kutta methods are both constant. The absolute stability regions of some methods are plotted and numerical examples show the efficiency of the method.展开更多
In this paper,we investigate the numerical performance of a family of P-stable two-step Maruyama schemes in mean-square sense for stochastic differential equations with time delay proposed in[8,10]for a certain class ...In this paper,we investigate the numerical performance of a family of P-stable two-step Maruyama schemes in mean-square sense for stochastic differential equations with time delay proposed in[8,10]for a certain class of nonlinear stochastic delay differential equations with multiplicative white noises.We also test the convergence of one of the schemes for a time-delayed Burgers’equation with an additive white noise.Numerical results show that this family of two-step Maruyama methods exhibit similar stability for nonlinear equations as that for linear equations.展开更多
文摘An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods （TSCRK） is presented. Their construction, their order conditions and their convergence are studied. The two-step continuity Runge-Kutta methods possess good numerical stability properties and higher stage-order, and keep the explicit process of computing the Runge-Kutta stages. The numerical experiments show that the TSCRK methods are efficient.
基金supported by the National Natural Science of China No.11971263,11871458Shandong Provincial Natural Science Foundation No.ZR2019ZD41National Key R&D Program of China No.2018YFA0703900。
文摘In this paper,a stochastic linear two-step scheme has been presented to approximate backward stochastic differential equations(BSDEs).A necessary and sufficient condition is given to judge the L 2-stability of our numerical schemes.This stochastic linear two-step method possesses a family of 3-order convergence schemes in the sense of strong stability.The coefficients in the numerical methods are inferred based on the constraints of strong stability and n-order accuracy(n∈N^(+)).Numerical experiments illustrate that the scheme is an efficient probabilistic numerical method.
基金Ministry of Education Humanities and Social Science Foundation Youth Project“Micro-quantification,Action Mechanism and Impact Research on Financialization of Entity Enterprises”[Grant number.19YJC790106]National Social Science Fund“Mechanism Analysis and Optimization Path Research of Digital Finance Supporting the Im‐provement of Development Efficiency of SMEs”[Grant number.21BJY047]+2 种基金Science and Technology Research Program of Chongqing Education Commission of China:“Optimization Path Research of Or‐ganizational Effectiveness of SOEs in Chongqing Based on Multi-Level Organizational Citizenship Behavior”[Grant number.17SKG036]Chongqing Social Science Planning Major Project“Research on the Technological Progress Path and Countermeasure System of Innovation-driven Manufacturing Upgrade in Chongqing”[Grant num‐ber.2020ZDJJ01]Chongqing Municipal Education Commission Hu‐manities and Social Sciences Research Project“Western Region Finan‐cial Development and Manufacturing Traditional Comparative Advan‐tage Transformation:Efficiency Measurement,Action Mechanism and Research on Spatial Effects”[Grant number.20SKGH040].
文摘The digital economy has become an important driver for stimulating economic growth.The digital economy has now widely penetrated the fields of economy and society,providing new opportunities for the develop‐ment of urban-rural integration.Based on panel data for 30 provinces in China from 2011 to 2020,this study constructed an index system for the integration of the digital economy and the development of urban-rural areas and conducted a systematic measurement analysis.Additionally,we used a two-step system of GMM estimation to analyze the impact of the digital economy on the development of urban-rural integra‐tion.The findings demonstrate the significant imbalance paradox of China’s digital economy development,which is shown in a gradient where the eastern region is higher than the center and the central region is higher than the west.Urban-rural integration levels in China fluctuate and display geographical variance,typically displaying high levels in the east and low levels in the west.Urban-rural integration is significantly encouraged by the digital economy,yet it varies in variability between different areas and dimensions.Addi‐tionally,rural human capital moderates the favorable effects of the digital economy on urban-rural integra‐tion.As a result,in order to achieve the integrated development of urban and rural areas,it is imperative to fully exploit the active role of the digital economy,better support the development of rural revitalization,bridge the“digital divide”between urban and rural development,and build a strong foundation for the for‐mation of a digital urban-rural integrated development pattern with urban and rural areas and common con‐struction and sharing.
文摘In this paper, a class of two-step continuity Runge-Kutta（TSCRK） methods for solving singular delay differential equations（DDEs） is presented. Analysis of numerical stability of this methods is given. We consider the two distinct cases： （i）τ≥ h, （ii）τ 〈 h, where the delay τ and step size h of the two-step continuity Runge-Kutta methods are both constant. The absolute stability regions of some methods are plotted and numerical examples show the efficiency of the method.
基金This work was supported by the NSF of China(No.10901036)and AIRFORCE MURI.The authors thank the referees for their helpful suggestions for improving the paper.The first author also would like to thank Professor George Em Karniadakis for his hospitality when she was visiting Division of Applied Mathematics at Brown University.
文摘In this paper,we investigate the numerical performance of a family of P-stable two-step Maruyama schemes in mean-square sense for stochastic differential equations with time delay proposed in[8,10]for a certain class of nonlinear stochastic delay differential equations with multiplicative white noises.We also test the convergence of one of the schemes for a time-delayed Burgers’equation with an additive white noise.Numerical results show that this family of two-step Maruyama methods exhibit similar stability for nonlinear equations as that for linear equations.
