{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:17:38Z","timestamp":1772252258006,"version":"3.50.1"},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,8,14]],"date-time":"2023-08-14T00:00:00Z","timestamp":1691971200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,8,14]],"date-time":"2023-08-14T00:00:00Z","timestamp":1691971200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100004052","name":"King Abdullah University of Science and Technology","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004052","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2023,10]]},"DOI":"10.1007\/s10915-023-02312-4","type":"journal-article","created":{"date-parts":[[2023,8,14]],"date-time":"2023-08-14T02:01:21Z","timestamp":1691978481000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Multiple-Relaxation Runge Kutta Methods for Conservative Dynamical Systems"],"prefix":"10.1007","volume":"97","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0741-6303","authenticated-orcid":false,"given":"Abhijit","family":"Biswas","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1212-126X","authenticated-orcid":false,"given":"David I.","family":"Ketcheson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,8,14]]},"reference":[{"issue":"2","key":"2312_CR1","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1006\/jcph.1997.5697","volume":"135","author":"A Arakawa","year":"1997","unstructured":"Arakawa, A.: Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow: part I. J. Comput. Phys. 135(2), 103\u2013114 (1997)","journal-title":"J. Comput. Phys."},{"key":"2312_CR2","doi-asserted-by":"crossref","unstructured":"Biswas, A., Ketcheson, D.I.: Code for multiple-relaxation Runge\u2013Kutta methods for conservative dynamical systems. https:\/\/github.com\/abhibsws\/Multiple_Relaxation_RK_Methods. (2023)","DOI":"10.1007\/s10915-023-02312-4"},{"issue":"12","key":"2312_CR3","doi-asserted-by":"publisher","first-page":"2646","DOI":"10.1016\/j.matcom.2011.05.007","volume":"81","author":"M Calvo","year":"2011","unstructured":"Calvo, M., Laburta, M., Montijano, J.I., R\u00e1ndez, L.: Error growth in the numerical integration of periodic orbits. Math. Comput. Simul. 81(12), 2646\u20132661 (2011)","journal-title":"Math. Comput. Simul."},{"issue":"3","key":"2312_CR4","doi-asserted-by":"publisher","first-page":"868","DOI":"10.1137\/04061979X","volume":"28","author":"MP Calvo","year":"2006","unstructured":"Calvo, M.P., Hern\u00e1ndez-Abreu, D., Montijano, J.I., R\u00e1ndez, L.: On the preservation of invariants by explicit Runge\u2013Kutta methods. SIAM J. Sci. Comput. 28(3), 868\u2013885 (2006)","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"2312_CR5","doi-asserted-by":"publisher","first-page":"936","DOI":"10.1137\/0914057","volume":"14","author":"MP Calvo","year":"1993","unstructured":"Calvo, M.P., Sanz-Serna, J.M.: The development of variable-step symplectic integrators, with application to the two-body problem. SIAM J. Sci. Comput. 14(4), 936\u2013952 (1993)","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"2312_CR6","doi-asserted-by":"publisher","first-page":"1391","DOI":"10.1137\/S0036142995281152","volume":"34","author":"B Cano","year":"1997","unstructured":"Cano, B., Sanz-Serna, J.M.: Error growth in the numerical integration of periodic orbits, with application to Hamiltonian and reversible systems. SIAM J. Numer. Anal. 34(4), 1391\u20131417 (1997)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"2312_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1093\/imanum\/7.1.1","volume":"7","author":"GJ Cooper","year":"1987","unstructured":"Cooper, G.J.: Stability of Runge\u2013Kutta methods for trajectory problems. IMA J. Numer. Anal. 7(1), 1\u201313 (1987)","journal-title":"IMA J. Numer. Anal."},{"issue":"4","key":"2312_CR8","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1007\/s002110050247","volume":"75","author":"J De Frutos","year":"1997","unstructured":"De Frutos, J., Sanz-Serna, J.M.: Accuracy and conservation properties in numerical integration: the case of the Korteweg-de Vries equation. Numer. Math. 75(4), 421\u2013445 (1997)","journal-title":"Numer. Math."},{"key":"2312_CR9","unstructured":"Dekker, K., Verwer, J.G.: Stability of Runge\u2013Kutta methods for stiff nonlinear differential equations. CWI Monographs 2 (1984)"},{"issue":"2","key":"2312_CR10","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1093\/imanum\/20.2.235","volume":"20","author":"A Dur\u00e1n","year":"2000","unstructured":"Dur\u00e1n, A., Sanz-Serna, J.M.: The numerical integration of relative equilibrium solutions: the nonlinear Schr\u00f6dinger equation. IMA J. Numer. Anal. 20(2), 235\u2013261 (2000)","journal-title":"IMA J. Numer. Anal."},{"key":"2312_CR11","unstructured":"Fehlberg, E.: Low-order classical Runge\u2013Kutta formulas with stepsize control and their application to some heat transfer problems, vol. 315. National Aeronautics and Space Administration (1969)"},{"issue":"1\u20134","key":"2312_CR12","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1016\/0167-2789(92)90246-J","volume":"60","author":"CW Gear","year":"1992","unstructured":"Gear, C.W.: Invariants and numerical methods for ODEs. Phys. D 60(1\u20134), 303\u2013310 (1992)","journal-title":"Phys. D"},{"key":"2312_CR13","unstructured":"Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations. Springer (2006)"},{"key":"2312_CR14","unstructured":"Heun, K.: Neue methoden zur approximativen integration der differentialgleichungen einer unabh\u00e4ngigen ver\u00e4nderlichen. Z. Math. Phys. 45, 23\u201338 (1900)"},{"issue":"5","key":"2312_CR15","doi-asserted-by":"publisher","first-page":"1","DOI":"10.9734\/BJMCS\/2015\/19270","volume":"10","author":"A Ionescu","year":"2015","unstructured":"Ionescu, A., Militaru, R., Munteanu, F.: Geometrical methods and numerical computations for prey-predator systems. Br. J. Math. Comput. Sci. 10(5), 1\u201315 (2015)","journal-title":"Br. J. Math. Comput. Sci."},{"issue":"1\u20132","key":"2312_CR16","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/S0168-9274(02)00138-1","volume":"44","author":"CA Kennedy","year":"2003","unstructured":"Kennedy, C.A., Carpenter, M.H.: Additive Runge\u2013Kutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math. 44(1\u20132), 139\u2013181 (2003)","journal-title":"Appl. Numer. Math."},{"key":"2312_CR17","doi-asserted-by":"crossref","unstructured":"Ketcheson, D.I.: Relaxation Runge\u2013Kutta methods: conservation and stability for inner-product norms. SIAM J. Numer. Anal. 57(6), 2850\u20132870 (2019)","DOI":"10.1137\/19M1263662"},{"issue":"6","key":"2312_CR18","doi-asserted-by":"publisher","first-page":"1839","DOI":"10.1137\/0732083","volume":"32","author":"S Li","year":"1995","unstructured":"Li, S., Vu-Quoc, L.: Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation. SIAM J. Numer. Anal. 32(6), 1839\u20131875 (1995)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"2312_CR19","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1016\/0771-050X(81)90010-3","volume":"7","author":"PJ Prince","year":"1981","unstructured":"Prince, P.J., Dormand, J.R.: High order embedded Runge\u2013Kutta formulae. J. Comput. Appl. Math. 7(1), 67\u201375 (1981)","journal-title":"J. Comput. Appl. Math."},{"key":"2312_CR20","doi-asserted-by":"crossref","unstructured":"Ranocha, H., Ketcheson, D.I.: Relaxation Runge\u2013Kutta methods for Hamiltonian problems. J. Sci. Comput. 84(1), 1\u201327 (2020)","DOI":"10.1007\/s10915-020-01277-y"},{"key":"2312_CR21","doi-asserted-by":"crossref","unstructured":"Ranocha, H., L\u00f3czi, L., Ketcheson, D.I.: General relaxation methods for initial-value problems with application to multistep schemes. Numer. Math. 146(4), 875\u2013906 (2020)","DOI":"10.1007\/s00211-020-01158-4"},{"key":"2312_CR22","doi-asserted-by":"crossref","unstructured":"Ranocha, H., Quezada de Luna, M., Ketcheson, D.I.: On the rate of error growth in time for numerical solutions of nonlinear dispersive wave equations. Part. Differ. Equ. Appl. 2(6), 1\u201326 (2021)","DOI":"10.1007\/s42985-021-00126-3"},{"key":"2312_CR23","doi-asserted-by":"crossref","unstructured":"Ranocha, H., Mitsotakis, D., Ketcheson, D.I.: A broad class of conservative numerical methods for dispersive wave equations. Commun. Comput. Phys. (2021)","DOI":"10.4208\/cicp.OA-2020-0119"},{"issue":"2","key":"2312_CR24","doi-asserted-by":"publisher","first-page":"A612","DOI":"10.1137\/19M1263480","volume":"42","author":"H Ranocha","year":"2020","unstructured":"Ranocha, H., Sayyari, M., Dalcin, L., Parsani, M., Ketcheson, D.I.: Relaxation Runge-Kutta methods: fully discrete explicit entropy-stable schemes for the compressible Euler and Navier-Stokes equations. SIAM J. Sci. Comput. 42(2), A612\u2013A638 (2020)","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"2312_CR25","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1016\/0021-9991(88)90177-5","volume":"77","author":"CW Shu","year":"1988","unstructured":"Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439\u2013471 (1988)","journal-title":"J. Comput. Phys."},{"key":"2312_CR26","volume-title":"Preservation of the invariants of Lotka-Volterra equations by iterated deferred correction methods","author":"M Uzunca","year":"2021","unstructured":"Uzunca, M.: Preservation of the invariants of Lotka-Volterra equations by iterated deferred correction methods. TWMS J. App. and Eng, Math (2021)"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-023-02312-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-023-02312-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-023-02312-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,23]],"date-time":"2023-09-23T11:07:00Z","timestamp":1695467220000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-023-02312-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,14]]},"references-count":26,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,10]]}},"alternative-id":["2312"],"URL":"https:\/\/doi.org\/10.1007\/s10915-023-02312-4","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,14]]},"assertion":[{"value":"10 February 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 July 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 July 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 August 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"On behalf of all authors, the corresponding author declares that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"4"}}