[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82694-en":3,"doc-seo-82694-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82694,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Nonlinear Parareal-Incomplete OSWR Method for the Coupled Reaction-Diffusion System: Convergence Analysis and Computational Strategies","A nonlinear extension of the Parareal-Incomplete Optimized Schwarz Waveform Relaxation (OSWR) method is developed to efficiently simulate coupled reaction-diffusion systems, with emphasis on models arising in cardiac electrophysiology. The approach combines spatial and temporal decomposition to address the computational burden of large-scale, nonlinear PDE simulations. A full convergence analysis accompanies the method, and extensive numerical experiments validate reliable convergence across diverse space-time subdomains, demonstrating robustness for complex multi-subdomain computations.","arXiv :2607 .02780v1 [math .NA] 2 Jul 2026  \nNonlinear Parareal-Incomplete OSWR Method for the Coupled Reaction-Diffusion System: Convergence Analysis and Computational  \nStrategies  \nGobinda Garai∗  \nInstitute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic  \nNagaiah Chamakuri†  \nSchool of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram,  \nTrivandrum, India  \nJuly 7, 2026  \nAbstract  \nThis paper presents a development of a nonlinear extension of the Parareal-Incomplete Optimized Schwarz Waveform Relaxation (OSWR) method, aimed at efficiently simulating coupled reactiondiffusion systems, with particular attention to those encountered in cardiac electrophysiology. The strategy leverages the synergy of spatial and temporal decomposition to tackle the computational challenges of large-scale, nonlinear simulations. The proposed approach is accompanied by a comprehensive convergence analysis. Extensive numerical experiments confirm the convergence of the nonlinear Parareal-Incomplete OSWR method. The results showcase its robust convergence behavior across various space-time subdomains, underscoring the method’s reliability and effectiveness, especially in complex simulations involving multiple subdomains.  \nKeywords: Coupled reaction-diffusion system, Parallel-in-Time (PinT), OSWR, Convergence analysis, Parareal method.  \nMSC Classification: 65M12, 65Y05, 65M15, 65Y20  \n1 Introduction  \nThe parareal-in-time algorithm has been extensively studied to enable parallel computations by dividing the temporal integration interval into smaller segments [21] . Similarly, optimized Schwarz waveform relaxation (OSWR) algorithms have been explored to address the computational challenges of solving large scale systems that arise from discretizing partial differential equations (PDEs) [12] . Our primary goal is to develop and refine parallel algorithms that combine the strengths of the parareal-intime and OSWR methods, working effectively across both spatial and temporal domains for nonlinear PDEs. This effort focuses on solving the complexities of nonlinear coupled reaction-diffusion systems, such as the Rogers-McCulloch (RM) model [30] . We denote a bounded connected domain by Ω ⊂ Rd , here d = 1 or 2, with Lipschitz continuous boundary ∂Ω . We represent the space-time domain and its lateral boundary are by Q = Ω× (0, T] and Σ = ∂Ω× (0, T], respectively. The RM model, constituting the focal point of our study, is governed by coupled reaction-diffusion equations given below.  \n ~~u~~t = γ∆u − c1 u (a − u)(1 − u) − c2 uv, (x, t) ∈ Q,  \n vtuν  b0u, − bc3 v, ((xx,, tt))  QΣ,, (1)  \n􀀾  \n􀀺 u(x, 0) = u0 , and v (x, 0) = v0 , x ∈ Ω ,  \nwhere ν represents the outward unit normal to ∂Ω . The problem parameters γ, a, b, c 1 , c2 and c3 appear in (1) are positive real numbers [30] . The RM model is a phenomenological two-variable model and helps to study cardiac action potential propagation within the myocardium, which is the muscular  \n∗ Email: [garai@math.cas.cz](garai@math.cas.cz)  \n†Email: [nagaiah.chamakuri@iisertvm.ac.in](nagaiah.chamakuri@iisertvm.ac.in)  \ntissue of the heart. The state variables u and v represent the spatial and temporal evolution of the transmembrane potential and the recovery variable, respectively. These models reduce the complex array of ion currents to two state variables that describe excitation and recovery through the cardiac tissue. Such kinds of models hold significance due to their ability to undergo thorough mathematical analysis of excitation and recovery processes, represented graphically in a phase plane where u is plotted against v [19] . Several modifications of the FitzHugh-Nagumo model have been proposed to mimic the rate sensitivity of action potential duration [1, 33 , 30 , 27] . Our investigation specifically centers on the RM model [30], notable for its maintenance of a stable resting potential in cardiac myocytes.  \ncoFormputbreviationt,yw, weecdaennotreaed","cbCair5VjP94eKMf","https://ap.wps.com/l/cbCair5VjP94eKMf","pdf",1455302,1,25,"English","en",105,"# Introduction\n## Problem setting and motivation\n## Time and space parallelization framework","[{\"question\":\"What problem does the nonlinear Parareal-Incomplete OSWR method target?\",\"answer\":\"It targets efficient simulation of coupled reaction-diffusion systems, particularly those relevant to cardiac electrophysiology, where nonlinearities and multi-scale behavior make computation difficult.\"},{\"question\":\"How does the proposed method introduce parallelism?\",\"answer\":\"It leverages the Parareal framework to split the time interval into subsegments and combines this with spatial decomposition through an OSWR-based strategy to work across space-time domains.\"},{\"question\":\"What evidence is provided for the method’s effectiveness?\",\"answer\":\"The document states that a comprehensive convergence analysis is provided and that extensive numerical experiments confirm convergence behavior across various space-time 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problem does the nonlinear Parareal-Incomplete OSWR method target?","Question",{"text":75,"@type":76},"It targets efficient simulation of coupled reaction-diffusion systems, particularly those relevant to cardiac electrophysiology, where nonlinearities and multi-scale behavior make computation difficult.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method introduce parallelism?",{"text":80,"@type":76},"It leverages the Parareal framework to split the time interval into subsegments and combines this with spatial decomposition through an OSWR-based strategy to work across space-time domains.",{"name":82,"@type":73,"acceptedAnswer":83},"What evidence is provided for the method’s effectiveness?",{"text":84,"@type":76},"The document states that a comprehensive convergence analysis is provided and that extensive numerical experiments confirm convergence behavior across various space-time 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