Fourth-Order Paired-Explicit Runge-Kutta Methods

Fourth-Order Paired-Explicit Runge-Kutta Methods | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Fourth-Order Paired-Explicit Runge-Kutta Methods Daniel Doehring, Lars Christmann, Michael Schlottke-Lakemper, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5259801/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 9 You are reading this latest preprint version Abstract In this paper, we extend the Paired-Explicit Runge-Kutta (PERK) schemes by Vermeire et. al. to fourth-order of consistency. Based on the order conditions for partitioned Runge-Kutta methods we motivate a specific form of the Butcher arrays which leads to a family of fourth-order accurate methods. The employed form of the Butcher arrays results in a special structure of the stability polynomials, which needs to be adhered to for an efficient optimization of the domain of absolute stability. We demonstrate that the constructed fourth-order PERK methods satisfy linear stability, internal consistency, designed order of convergence, and conservation of linear invariants. At the same time, these schemes are seamlessly coupled for codes employing a method-of-lines approach, in particular without any modifications of the spatial discretization. We demonstrate speedup for single-threaded program executions, shared-memory parallelism, i.e., multi-threaded executions and distributed-memory parallelism with MPI. We apply the multirate PERK schemes to inviscid and viscous problems with locally varying wave speeds, which may be induced by non-uniform grids or multiscale properties of the governing partial differential equation. Compared to state-of-the-art optimized standalone methods, the multirate PERK schemes allow significant reductions in right-hand-side evaluations and wall-clock time, ranging from 66% up to factors greater than four. A reproducibility repository is provided which enables the reader to examine all results presented in this work. Multirate Time Integration Runge-Kutta Methods Method of Lines High Order 2008 MSC: 65L06 65M20 7604 Full Text Additional Declarations No competing interests reported. Supplementary Files SupplementaryMaterial.zip Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 25 Aug, 2025 Reviews received at journal 18 May, 2025 Reviews received at journal 16 May, 2025 Reviewers agreed at journal 05 Apr, 2025 Reviewers agreed at journal 04 Apr, 2025 Reviewers agreed at journal 31 Mar, 2025 Reviewers invited by journal 31 Mar, 2025 Submission checks completed at journal 27 Mar, 2025 First submitted to journal 25 Mar, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-5259801","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":436249663,"identity":"1c47bf54-480d-47dc-bb15-67a164a78556","order_by":0,"name":"Daniel Doehring","email":"data:image/png;base64,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","orcid":"","institution":"RWTH Aachen University","correspondingAuthor":true,"prefix":"","firstName":"Daniel","middleName":"","lastName":"Doehring","suffix":""},{"id":436249664,"identity":"bbdf4300-ddf7-4922-881b-99d675896ce8","order_by":1,"name":"Lars Christmann","email":"","orcid":"","institution":"University of Cologne","correspondingAuthor":false,"prefix":"","firstName":"Lars","middleName":"","lastName":"Christmann","suffix":""},{"id":436249666,"identity":"f5217712-7f7a-42c9-99e8-06d619fddbab","order_by":2,"name":"Michael Schlottke-Lakemper","email":"","orcid":"","institution":"University of Augsburg","correspondingAuthor":false,"prefix":"","firstName":"Michael","middleName":"","lastName":"Schlottke-Lakemper","suffix":""},{"id":436249668,"identity":"4a2cb4aa-6978-420e-af4d-65020398a700","order_by":3,"name":"Gregor J. 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Based on the order conditions for partitioned Runge-Kutta methods we motivate a specific form of the Butcher arrays which leads to a family of fourth-order accurate methods. The employed form of the Butcher arrays results in a special structure of the stability polynomials, which needs to be adhered to for an efficient optimization of the domain of absolute stability.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eWe demonstrate that the constructed fourth-order PERK methods satisfy linear stability, internal consistency, designed order of convergence, and conservation of linear invariants. At the same time, these schemes are seamlessly coupled for codes employing a method-of-lines approach, in particular without any modifications of the spatial discretization. We demonstrate speedup for single-threaded program executions, shared-memory parallelism, i.e., multi-threaded executions and distributed-memory parallelism with MPI.\u003c/p\u003e\n\u003cp\u003eWe apply the multirate PERK schemes to inviscid and viscous problems with locally varying wave speeds, which may be induced by non-uniform grids or multiscale properties of the governing partial differential equation. 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