[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83481-en":3,"doc-seo-83481-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83481,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782698725881665579",8,"Research & Report","Exponential Low-Regularity Parareal Algorithms for Nonlinear Schrödinger Equations","The parareal algorithm is a widely studied parallel-in-time method for approximating time-dependent problems. For non-diffusive dynamics, standard parareal schemes may converge slowly or become unstable because phase errors across Fourier modes can be amplified. This work studies the nonlinear Schrödinger equation (NLS) as a representative case, developing a convergence framework for limited-regularity solutions under stability and local truncation error assumptions on the coarse propagator. Exponential low-regularity integrators are validated for 1D quadratic and cubic NLS, enabling provably linearly convergent parareal algorithms with contraction proportional to the coarse time step. Numerical results for quadratic, cubic, and quintic NLS show rapid convergence and improved performance over classical coarse propagators.","arXiv :2607 .00384v1 [math .NA] 1 Jul 2026  \nExponential Low-Regularity Parareal Algorithms for  \nNonlinear Schrdinger Equations *  \nQingle Lin† Zhi Zhou†  \nJuly 2, 2026  \nAbstract  \nThe parareal algorithm is one of the most widely studied parallel-in-time methods for the numerical approximation of time-dependent problems. For non-diffusive equations, however, standard parareal methods may converge slowly or even become unstable due to the absence of damping, while nonlinear interactions can transfer and amplify phase errors across Fourier modes. In this work, we consider the nonlinear Schrdinger equation (NLS) as a representative non-diffusive model and analyze parareal algorithms with an exact fine propagator, with particular emphasis on the design of suitable coarse propagators. We establish a general convergence framework, valid for solutions with limited regularity, under stability and local truncation error assumptions on the coarse propagator. These assumptions are verified for selected exponential low-regularity integrators designed for one-dimensional quadratic and cubic NLS equations, which achieve optimal approximation orders without derivative loss. To the best of our knowledge, this is the first construction of parareal algorithms for NLS equations that are provably linearly convergent, with a contraction factor proportional to the coarse time-step size even for solutions of limited regularity. Numerical experiments on quadratic, cubic, and quintic NLS equations demonstrate rapid convergence and improved performance over parareal variants using classical coarse propagators, including Lie and Strang splitting methods and first-and third-order exponential Runge–Kutta integrators.  \nKeywords: nonlinear Schrdinger equation; parareal algorithm; parallel-in-time integration; coarse propagator; exponential low-regularity integrator; convergence analysis  \nAMS subject classifications: 65M12, 65Y05  \n1 Introduction  \nIn this paper, we consider the parallel-in-time (PinT) numerical approximation of the nonlinear Schrdinger equation (NLS)  \ni∂tu = −∆u + F(u, u) , (t, x) ∈ (0, T) × Td , (1 . 1)  \nsubject to the initial condition u(0) = u0 ∈ Hr(Td ) . Here, ∆ = Pdi=1 ∂2xi and Td = (0 , 2π)d denotes the ddimensional torus, corresponding to periodic boundary conditions. NLS-type models arise in many areas of physics, including nonlinear optics, Bose–Einstein condensates, deep water waves, and plasma physics [5, 11, 26, 27, 43] . The numerical approximation of (1.1) is typically carried out using sequential time-stepping schemes, which advance the solution incrementally in time. This inherently sequential structure constitutes  \n*This work is partially supported by National Natural Science Foundation of China (Project 12422117 and Project 12426312), Hong Kong Research Grants Council (Project 15302323) and an internal grant of Hong Kong Polytechnic University (Project P0053938, Work Programme: 4-ZZVA) .  \n†Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, P.R. China ([qingle.lin@connect.polyu.hk](qingle.lin@connect.polyu.hk) , [zhizhou@polyu.edu.hk](zhizhou@polyu.edu.hk))  \na major computational bottleneck, particularly for long-time simulations. Moreover, as processor clock speeds have approached fundamental physical limits, further improvements in computational performance must increasingly rely on greater parallelism through the use of larger numbers of cores [25] .  \nPinT methods have been an active area of research for more than two decades in the numerical solution of large-scale evolution problems, particularly when spatial parallelism becomes saturated. The earliest pioneering work can be traced back to Nievergelt in 1964 [38] . Several well-established PinT algorithms, such as the parareal method [23, 35], multigrid reduction in time (MGRIT) [14, 16], parallel full approximation scheme in space and time (PFASST) [15, 37], diagonalization technique [3, 24, 51], and Laplace transfo","cbCaiswWesasvNwC","https://ap.wps.com/l/cbCaiswWesasvNwC","pdf",969691,3,1,26,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"Why do standard parareal methods struggle for non-diffusive equations like the nonlinear Schrödinger equation?\",\"answer\":\"They may converge slowly or become unstable because the lack of damping allows phase errors to be transferred and amplified across Fourier modes, undermining correction quality across iterations.\"},{\"question\":\"What is the key idea behind designing improved parareal algorithms in this work?\",\"answer\":\"The coarse propagator is carefully constructed using exponential low-regularity integrators, and the paper establishes assumptions on stability and local truncation error to enable a general convergence framework.\"},{\"question\":\"What convergence behavior is proven for the constructed NLS parareal algorithms?\",\"answer\":\"The paper claims provable linear convergence, with a contraction factor proportional to the coarse time-step size even for solutions with limited regularity, supported by analyzed integrators and numerical experiments.\"}]",1784188315,66,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"exponential-low-regularity-parareal-algorithms-for-nonlinear-schrodinger-equations","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/exponential-low-regularity-parareal-algorithms-for-nonlinear-schrodinger-equations/83481/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why do standard parareal methods struggle for non-diffusive equations like the nonlinear Schrödinger equation?","Question",{"text":75,"@type":76},"They may converge slowly or become unstable because the lack of damping allows phase errors to be transferred and amplified across Fourier modes, undermining correction quality across iterations.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the key idea behind designing improved parareal algorithms in this work?",{"text":80,"@type":76},"The coarse propagator is carefully constructed using exponential low-regularity integrators, and the paper establishes assumptions on stability and local truncation error to enable a general convergence framework.",{"name":82,"@type":73,"acceptedAnswer":83},"What convergence behavior is proven for the constructed NLS parareal algorithms?",{"text":84,"@type":76},"The paper claims provable linear convergence, with a contraction factor proportional to the coarse time-step size even for solutions with limited regularity, supported by analyzed integrators and numerical experiments.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & 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