[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83710-en":3,"doc-seo-83710-105":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},83710,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Convergence of Substructuring Waveform Relaxation Algorithms for Hyperbolic PDEs with Time Delay","The study investigates Dirichlet-Neumann Waveform Relaxation (DNWR) and Neumann-Neumann Waveform Relaxation (NNWR) for solving hyperbolic partial differential equations with time delay. It focuses on stability, convergence, and computational efficiency of non-overlapping substructuring methods using asymmetric domain decomposition. Fourier analysis yields linear convergence estimates for numerical errors, while Laplace transform analysis characterizes finite-step convergence. Optimal parameters are derived for heterogeneous spatial domains, and numerical experiments support the theory by improving accuracy while reducing computational complexity.","Convergence of Substructuring Waveform Relaxation Algorithms for Hyperbolic PDEs with Time Delay  \nBankim C. Mandala , Deeksha Tomera,∗  \na School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Odisha, 752050, India  \n3 Jul 2026  \nARTICLE INFO  \nKeywords:  \nWaveform Relaxation  \nAsymmetric Domain Decomposition Hyperbolic PDE with Time Delay Dirichlet-Neumann  \nNeumann-Neumann  \nABSTRACT  \nThis article investigates the application and analysis of two substructuring waveform relaxation algorithms namely Dirichlet-Neumann Waveform Relaxation (DNWR) and Neumann-Neumann Waveform Relaxation (NNWR) for solving hyperbolic partial differential equations (PDEs) with time delay. These equations are relevant in numerous physical and engineering contexts, such as wave propagation, biological processes, and control systems, where the system’s dynamics are influenced by past states. The study emphasizes the stability, convergence, and computational efficiency of these non-overlapping domain decomposition methods when applied to such problems. Specifically, the DNWR and NNWR algorithms are analyzed using both Fourier and Laplace transforms in asymmetric domain decomposition to assess their capability to manage delayed terms in hyperbolic systems. Using Fourier analysis, we establish linear convergence estimate for the numerical errors. Laplace transform analysis enables a more in-depth study for characterizing finite-step convergence. Additionally, we derive the optimal parameters required  \narXiv :2607 .03326v1 [math .NA]  \nto achieve finite step convergence in presence of heterogeneous spatial domain. Theoretical findings are complemented by numerical experiments, showcasing the methods’ effectiveness in maintaining accuracy while reducing computational complexity. Additionally, the study explores potential extensions to more complex problems and diverse applications.  \n1. Introduction  \nHyperbolic PDEs with time delay are fundamental mathematical models used to describe dynamic systems in which the systems evolution is influenced by both its current state as well as its previous states. Such equations arise across abroad range of scientific and engineering problems, including wave propagation in viscoelastic materials (Hale, 1977), control systems with delayed feedback (Gu, Chen and Kharitonov, 2003), and biological processes such as population dynamics and neural networks (Cooke, 1963) . The inclusion of time delays in these systems introduces additional complexity, making their analysis and numerical solution both challenging and essential for understanding real-world phenomena.  \nDomain decomposition methods (DDMs) are powerful parallel computational tools for solving large-scale PDEs, achieved by partitioning the computational domain into a collection of smaller, more manageable subdomains (Quarteroni and Valli, 1999) . DDMs are generally classified into two categories: overlapping and non-overlapping. Non-overlapping DDMs are often referred to as substructuring methods. When applied to time-dependent problems, these are known as substructuring waveform relaxation methods. Among them, Dirichlet-Neumann Waveform Relaxation (DNWR) and Neumann-Neumann Waveform Relaxation (NNWR) have gained significant attention due to their efficiency in handling both linear and non-linear time-dependent problems (Garai and Mandal, 2023; Sana and Mandal, 2023) . These methods iteratively solve subproblems on non-overlapping subdomains by imposing appropriate  \nboundary conditions, such as Dirichlet or Neumann conditions, at the interfaces. While DNWR and NNWR have been extensively studied for parabolic and hyperbolic PDEs (Gander, Kwok and Mandal, 2016b,a) their application to PDEs with time delay remains relatively unexplored, despite the growing interest in delay-dependent systems (Michiels and Niculescu, 2007; Zhong, 2006) .  \nAnother popular version of domain decomposition methods, Schwarz Waveform Relaxation (SWR) method (Gander and Stuart, 1998; Ga","cbCaihYxRckLQs25","https://ap.wps.com/l/cbCaihYxRckLQs25","pdf",2095989,3,1,24,"English","en",105,"# Introduction\n## Hyperbolic PDEs with Time Delay and Applications\n## Domain Decomposition and Waveform Relaxation Methods\n## Schwarz Waveform Relaxation and Parallel Scalability\n## Main Contributions","[{\"question\":\"What problem do the DNWR and NNWR algorithms address in this study?\",\"answer\":\"They address solving hyperbolic partial differential equations that include time-delay terms, where the dynamics depend on both current and past states.\"},{\"question\":\"How is convergence analyzed for DNWR and NNWR?\",\"answer\":\"DNWR linear convergence for numerical errors is established via Fourier analysis, while Laplace transform analysis is used to characterize finite-step convergence; NNWR finite-step convergence is also proved.\"},{\"question\":\"What additional result is derived for DNWR in heterogeneous spatial domains?\",\"answer\":\"The study derives the optimal parameters needed to achieve finite-step convergence when the spatial domain is heterogeneous, and validates effectiveness with numerical experiments.\"}]",1784189906,60,{"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},"convergence-of-substructuring-waveform-relaxation-algorithms-for-hyperbolic-pdes-with-time-delay","",{"@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/convergence-of-substructuring-waveform-relaxation-algorithms-for-hyperbolic-pdes-with-time-delay/83710/",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","2026-07-27","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},"What problem do the DNWR and NNWR algorithms address in this study?","Question",{"text":75,"@type":76},"They address solving hyperbolic partial differential equations that include time-delay terms, where the dynamics depend on both current and past states.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is convergence analyzed for DNWR and NNWR?",{"text":80,"@type":76},"DNWR linear convergence for numerical errors is established via Fourier analysis, while Laplace transform analysis is used to characterize finite-step convergence; NNWR finite-step convergence is also proved.",{"name":82,"@type":73,"acceptedAnswer":83},"What additional result is derived for DNWR in heterogeneous spatial domains?",{"text":84,"@type":76},"The study derives the optimal parameters needed to achieve finite-step convergence when the spatial domain is heterogeneous, and validates effectiveness with 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},[93,97,101,105,109,114,119,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":29,"slug":108},5,"Comic","comic",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]