Pub Date : 2023-06-01DOI: 10.4208/jcm.2211-m2021-0293
Dingwen Li
Du Fort-Frankel finite difference method (FDM) was firstly proposed for linear diffu-sion equations with periodic boundary conditions by Du Fort and Frankel in 1953. It is an explicit and unconditionally von Neumann stable scheme. However, there has been no research work on numerical solutions of nonlinear Schr¨odinger equations with wave operator by using Du Fort-Frankel-type finite difference methods (FDMs). In this study, a class of invariants-preserving Du Fort-Frankel-type FDMs are firstly proposed for one-dimensional (1D) and two-dimensional (2D) nonlinear Schr¨odinger equations with wave operator. By using the discrete energy method, it is shown that their solutions possess the discrete energy and mass conservative laws, and conditionally converge to exact solutions with an order of O ( τ 2 + h 2 x +( τ/h x ) 2 ) for 1D problem and an order of O ( τ 2 + h 2 x + h 2 y +( τ/h x ) 2 +( τ/h y ) 2 ) for 2D problem in H 1 -norm. Here, τ denotes time-step size, while, h x and h y represent spatial meshsizes in x - and y -directions, respectively. Then, by introducing a stabilized term, a type of stabilized invariants-preserving Du Fort-Frankel-type FDMs are devised. They not only preserve the discrete energies and masses, but also own much better stability than original schemes. Finally, numerical results demonstrate the theoretical analyses.
{"title":"Invariants-Preserving Du Fort-Frankel Schemes and Their Analyses for Nonlinear Schrödinger Equations With Wave Operator","authors":"Dingwen Li","doi":"10.4208/jcm.2211-m2021-0293","DOIUrl":"https://doi.org/10.4208/jcm.2211-m2021-0293","url":null,"abstract":"Du Fort-Frankel finite difference method (FDM) was firstly proposed for linear diffu-sion equations with periodic boundary conditions by Du Fort and Frankel in 1953. It is an explicit and unconditionally von Neumann stable scheme. However, there has been no research work on numerical solutions of nonlinear Schr¨odinger equations with wave operator by using Du Fort-Frankel-type finite difference methods (FDMs). In this study, a class of invariants-preserving Du Fort-Frankel-type FDMs are firstly proposed for one-dimensional (1D) and two-dimensional (2D) nonlinear Schr¨odinger equations with wave operator. By using the discrete energy method, it is shown that their solutions possess the discrete energy and mass conservative laws, and conditionally converge to exact solutions with an order of O ( τ 2 + h 2 x +( τ/h x ) 2 ) for 1D problem and an order of O ( τ 2 + h 2 x + h 2 y +( τ/h x ) 2 +( τ/h y ) 2 ) for 2D problem in H 1 -norm. Here, τ denotes time-step size, while, h x and h y represent spatial meshsizes in x - and y -directions, respectively. Then, by introducing a stabilized term, a type of stabilized invariants-preserving Du Fort-Frankel-type FDMs are devised. They not only preserve the discrete energies and masses, but also own much better stability than original schemes. Finally, numerical results demonstrate the theoretical analyses.","PeriodicalId":50225,"journal":{"name":"Journal of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44548979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.4208/nmtma.oa-2022-0202
Mingguang Geng and Shuli Sun
{"title":"Projection Improved SPAI Preconditioner for FGMRES","authors":"Mingguang Geng and Shuli Sun","doi":"10.4208/nmtma.oa-2022-0202","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2022-0202","url":null,"abstract":"","PeriodicalId":50225,"journal":{"name":"Journal of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135195042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.4208/nmtma.oa-2022-0185
Xu Qian, Hong Zhang, Jingye Yan and Songhe Song
{"title":"Novel High-Order Mass- and Energy-Conservative Runge-Kutta Integrators for the Regularized Logarithmic Schrödinger Equation","authors":"Xu Qian, Hong Zhang, Jingye Yan and Songhe Song","doi":"10.4208/nmtma.oa-2022-0185","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2022-0185","url":null,"abstract":"","PeriodicalId":50225,"journal":{"name":"Journal of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135144382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.4208/nmtma.oa-2023-0044
Ruisong Gao, Yufeng Wang, Min Yang and Chuanjun Chen
{"title":"PI-VEGAN: Physics Informed Variational Embedding Generative Adversarial Networks for Stochastic Differential Equations","authors":"Ruisong Gao, Yufeng Wang, Min Yang and Chuanjun Chen","doi":"10.4208/nmtma.oa-2023-0044","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2023-0044","url":null,"abstract":"","PeriodicalId":50225,"journal":{"name":"Journal of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135095882","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.4208/jcm.2301-m2022-0208
Chunxiong Zheng, Xianwei Wen, Jinyu Zhang and Zhenya Zhou
Asymptotic theory for the circuit envelope analysis is developed in this paper. A typical feature of circuit envelope analysis is the existence of two significantly distinct timescales: one is the fast timescale of carrier wave, and the other is the slow timescale of modulation signal. We first perform pro forma asymptotic analysis for both the driven and autonomous systems. Then resorting to the Floquet theory of periodic operators, we make a rigorous justification for first-order asymptotic approximations. It turns out that these asymptotic results are valid at least on the slow timescale. To speed up the computation of asymptotic approximations, we propose a periodization technique, which renders the possibility of utilizing the NUFFT algorithm. Numerical experiments are presented, and the results validate the theoretical findings.
{"title":"Asymptotic Theory for the Circuit Envelope Analysis","authors":"Chunxiong Zheng, Xianwei Wen, Jinyu Zhang and Zhenya Zhou","doi":"10.4208/jcm.2301-m2022-0208","DOIUrl":"https://doi.org/10.4208/jcm.2301-m2022-0208","url":null,"abstract":"Asymptotic theory for the circuit envelope analysis is developed in this paper. A typical feature of circuit envelope analysis is the existence of two significantly distinct timescales: one is the fast timescale of carrier wave, and the other is the slow timescale of modulation signal. We first perform pro forma asymptotic analysis for both the driven and autonomous systems. Then resorting to the Floquet theory of periodic operators, we make a rigorous justification for first-order asymptotic approximations. It turns out that these asymptotic results are valid at least on the slow timescale. To speed up the computation of asymptotic approximations, we propose a periodization technique, which renders the possibility of utilizing the NUFFT algorithm. Numerical experiments are presented, and the results validate the theoretical findings.","PeriodicalId":50225,"journal":{"name":"Journal of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47375271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.4208/nmtma.oa-2023-0039
Teng-Teng Yao, Xiao-Qing Jin and Zhi Zhao
{"title":"A Riemannian Inertial Mann Algorithm for Nonexpansive Mappings on Hadamard Manifolds","authors":"Teng-Teng Yao, Xiao-Qing Jin and Zhi Zhao","doi":"10.4208/nmtma.oa-2023-0039","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2023-0039","url":null,"abstract":"","PeriodicalId":50225,"journal":{"name":"Journal of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135144242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.4208/nmtma.oa-2022-0190
Yuchao Tang, Shirong Deng and Tieyong Zeng
{"title":"Low Rank and Total Variation Based Two-Phase Method for Image Deblurring with Salt-and-Pepper Impulse Noise","authors":"Yuchao Tang, Shirong Deng and Tieyong Zeng","doi":"10.4208/nmtma.oa-2022-0190","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2022-0190","url":null,"abstract":"","PeriodicalId":50225,"journal":{"name":"Journal of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135145124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.4208/jcm.2207-m2022-0020
Hong-lin Liao, Tao Tang null, Tao Zhou
{"title":"Discrete Energy Analysis of the Third-Order Variable-Step BDF Time-Stepping for Diffusion Equations","authors":"Hong-lin Liao, Tao Tang null, Tao Zhou","doi":"10.4208/jcm.2207-m2022-0020","DOIUrl":"https://doi.org/10.4208/jcm.2207-m2022-0020","url":null,"abstract":"","PeriodicalId":50225,"journal":{"name":"Journal of Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136177614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：