{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T09:37:00Z","timestamp":1770975420237,"version":"3.50.1"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T00:00:00Z","timestamp":1769126400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T00:00:00Z","timestamp":1769126400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100021856","name":"Ministero dell\u2019Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["PRIN PNRR 2022 project BAT MEN (P20228C2PP, CUP: F53D23010020001)"],"award-info":[{"award-number":["PRIN PNRR 2022 project BAT MEN (P20228C2PP, CUP: F53D23010020001)"]}],"id":[{"id":"10.13039\/501100021856","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100021856","name":"Ministero dell\u2019Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["PRIN PNRR 2022 project MatForPat (P2022WC2ZZ, CUP: E53D23018030001)"],"award-info":[{"award-number":["PRIN PNRR 2022 project MatForPat (P2022WC2ZZ, CUP: E53D23018030001)"]}],"id":[{"id":"10.13039\/501100021856","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100017142","name":"Gruppo Nazionale per il Calcolo Scientifico","doi-asserted-by":"publisher","award":["GNCS\u2013INdAM"],"award-info":[{"award-number":["GNCS\u2013INdAM"]}],"id":[{"id":"10.13039\/100017142","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004837","name":"Ministerio de Ciencia e Innovaci\u00f3n","doi-asserted-by":"publisher","award":["PID2022\u2013141385NB\u2013I00"],"award-info":[{"award-number":["PID2022\u2013141385NB\u2013I00"]}],"id":[{"id":"10.13039\/501100004837","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":[[2026,2]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>The class of Time Accurate and highly Stable Explicit Runge\u2013Kutta (RK\u2013TASE) methods has been introduced by Bassenne, Fu and Mani (J. Comput. Phys. 2021) and then extended by Calvo, Montijano and R\u00e1ndez (J. Comput. Phys. 2021), and Aceto, Conte and Pagano (Appl. Numer. Math. 2024), for the efficient solution of stiff initial value problems. With the aim of making RK\u2013TASE methods suitable for the efficient solution of semi\u2013discretized reaction\u2013diffusion Partial Differential Equations (PDEs), in this work we exploit a general formulation of the schemes that allows to reduce both their computational cost and error constants, obtaining also good stability properties for such problems. In particular, a thorough study of the accuracy and stability properties leads to new RK\u2013TASE methods up to order five that are more efficient than existing ones. Several experiments show that the new proposed RK\u2013TASE methods are able to efficiently solve models of PDEs from applications that require integration over long time intervals. Furthermore, they show the better performance of the new RK\u2013TASE compared to other numerical schemes from the scientific literature.<\/jats:p>","DOI":"10.1007\/s10915-026-03184-0","type":"journal-article","created":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T03:33:58Z","timestamp":1769139238000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["General Runge\u2013Kutta TASE Methods for Reaction\u2013Diffusion Problems"],"prefix":"10.1007","volume":"106","author":[{"given":"Dajana","family":"Conte","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Juan Ignacio","family":"Montijano","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6373-9473","authenticated-orcid":false,"given":"Giovanni","family":"Pagano","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Beatrice","family":"Paternoster","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luis","family":"R\u00e1ndez","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,1,23]]},"reference":[{"key":"3184_CR1","doi-asserted-by":"crossref","unstructured":"Aceto, L., Conte, D., Pagano, G.: On a generalization of time-accurate and highly-stable explicit operators for stiff problems. Appl. Numer. Math. 200, 2\u201317 (2024)","DOI":"10.1016\/j.apnum.2023.04.001"},{"key":"3184_CR2","doi-asserted-by":"crossref","unstructured":"Aceto, L., Conte, D., Pagano, G.: Modified TASE Runge-Kutta methods for integrating stiff differential equations. SIAM J. Sci. Comput., 47(3), (2025)","DOI":"10.1137\/24M1667336"},{"key":"3184_CR3","doi-asserted-by":"crossref","unstructured":"Bassenne, M., Fu, L., Mani, A.: Time-accurate and highly-stable explicit operators for stiff differential equations. J. Comput. Phys., 424, (2021)","DOI":"10.1016\/j.jcp.2020.109847"},{"key":"3184_CR4","doi-asserted-by":"publisher","DOI":"10.1002\/9780470753767","volume-title":"Numerical methods for ordinary differential equations","author":"JC Butcher","year":"2008","unstructured":"Butcher, J.C.: Numerical methods for ordinary differential equations, 2nd edn. John Wiley & Sons Ltd, Chichester (2008)","edition":"2"},{"key":"3184_CR5","unstructured":"Caliari, M., Cassini, F.: Direction splitting of $$\\varphi $$-functions in exponential integrators for d-dimensional problems in Kronecker form. J. Approx. Softw., 1(1), (2024)"},{"key":"3184_CR6","doi-asserted-by":"crossref","unstructured":"Caliari, M., Cassini, F.: A second order directional split exponential integrator for systems of advection\u2013diffusion\u2013reaction equations. J. Comput. Phys., 498:112640, (2024)","DOI":"10.1016\/j.jcp.2023.112640"},{"key":"3184_CR7","doi-asserted-by":"crossref","unstructured":"Calvo, M., Fu, L., Montijano, J. I., R\u00e1ndez, L.: Singly TASE operators for the numerical solution of stiff differential equations by explicit Runge\u2013Kutta schemes. J. Sci. Comput., 96(1), (2023)","DOI":"10.1007\/s10915-023-02232-3"},{"key":"3184_CR8","doi-asserted-by":"crossref","unstructured":"M. Calvo, S. Gonz\u00e1lez-Pinto, J. I. Montijano. Runge-Kutta methods for the numerical solution of stiff semilinear systems. BIT Numer. Math., 40:611\u2013639, (2000)","DOI":"10.1023\/A:1022332200092"},{"key":"3184_CR9","doi-asserted-by":"crossref","unstructured":"Calvo, M., Montijano, J. I., R\u00e1ndez, L.: A note on the stability of time\u2013accurate and highly\u2013stable explicit operators for stiff differential equations. J. Comput. Phys., 436, (2021)","DOI":"10.1016\/j.jcp.2021.110316"},{"key":"3184_CR10","doi-asserted-by":"crossref","unstructured":"Calvo, M., Montijano, J. I., R\u00e1ndez, L.: Modified Singly-Runge\u2013Kutta-TASE methods for the numerical solution of stiff differential equations. J. Sci. Comput., 103(1), (2025)","DOI":"10.1007\/s10915-025-02813-4"},{"issue":"2","key":"3184_CR11","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1007\/s10915-024-02579-1","volume":"100","author":"D Conte","year":"2024","unstructured":"Conte, D., Gonz\u00e1lez-Pinto, S., Hern\u00e1ndez-Abreu, D., Pagano, G.: On Approximate Matrix Factorization and TASE W-methods for the Time Integration of Parabolic Partial Differential Equations. J. Sci. Comput. 100(2), 34 (2024)","journal-title":"J. Sci. Comput."},{"key":"3184_CR12","doi-asserted-by":"crossref","unstructured":"Conte, D., Martin-Vaquero, J., Pagano, G., Paternoster, B.: Stability theory of TASE\u2013Runge\u2013Kutta methods with inexact Jacobian. SIAM J. Sci. Comput. 46, A3638\u2013A3657 (2024)","DOI":"10.1137\/24M1631869"},{"key":"3184_CR13","doi-asserted-by":"crossref","unstructured":"Conte, D., Pagano, G., Paternoster, B.: Nonstandard finite differences numerical methods for a vegetation reaction\u2013diffusion model. J. Comput. Appl. Math., 419, (2023)","DOI":"10.1016\/j.cam.2022.114790"},{"key":"3184_CR14","doi-asserted-by":"crossref","unstructured":"Conte, D., Pagano, G., Paternoster, B.: Time-accurate and highly-stable explicit peer methods for stiff differential problems. Comm. Nonlinear Sci. Numer. Simul., 119, (2023)","DOI":"10.1016\/j.cnsns.2023.107136"},{"key":"3184_CR15","doi-asserted-by":"publisher","first-page":"2290","DOI":"10.1007\/s11538-019-00606-z","volume":"81","author":"L Eigentler","year":"2019","unstructured":"Eigentler, L., Sherratt, J.A.: Metastability as a coexistence mechanism in a model for dryland vegetation patterns. Bull. Math. Biol. 81, 2290\u20132322 (2019)","journal-title":"Bull. Math. Biol."},{"key":"3184_CR16","doi-asserted-by":"crossref","unstructured":"Frasca-Caccia, G., Valentino, C., Colace, F., Conte, D.: An overview of differential models for corrosion of cultural heritage artefacts. Math. Model. Nat. Phenom., 18, (2023)","DOI":"10.1051\/mmnp\/2023031"},{"key":"3184_CR17","doi-asserted-by":"publisher","first-page":"286","DOI":"10.1016\/j.apnum.2023.07.010","volume":"200","author":"M Frittelli","year":"2024","unstructured":"Frittelli, M., Sgura, I.: Matrix-oriented FEM formulation for reaction-diffusion PDEs on a large class of 2D domains. Appl. Numer. Math. 200, 286\u2013308 (2024)","journal-title":"Appl. Numer. Math."},{"issue":"2","key":"3184_CR18","doi-asserted-by":"publisher","first-page":"363","DOI":"10.3934\/mine.2024015","volume":"6","author":"M Frittelli","year":"2024","unstructured":"Frittelli, M., Sgura, I., Bozzini, B.: Turing patterns in a 3D morpho-chemical bulk-surface reaction-diffusion system for battery modeling. Math. Eng. 6(2), 363\u2013393 (2024)","journal-title":"Math. Eng."},{"key":"3184_CR19","volume-title":"The theory of matrices","author":"FR Gantmacher","year":"1998","unstructured":"Gantmacher, F.R.: The theory of matrices. AMS Chelsea Publishing, Providence, RI, first edition (1998)"},{"key":"3184_CR20","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1016\/j.apnum.2023.03.007","volume":"188","author":"S Gonz\u00e1lez-Pinto","year":"2023","unstructured":"Gonz\u00e1lez-Pinto, S., Hern\u00e1ndez-Abreu, D., Pagano, G., P\u00e9rez-Rodr\u00edguez, S.: Generalized TASE RK methods for stiff problems. Appl. Numer. Math. 188, 129\u2013145 (2023)","journal-title":"Appl. Numer. Math."},{"key":"3184_CR21","doi-asserted-by":"crossref","unstructured":"Hairer, E., Wanner, G.: Solving ordinary differential equations II. Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics (14). Springer, Berlin, (1996)","DOI":"10.1007\/978-3-642-05221-7"},{"key":"3184_CR22","doi-asserted-by":"crossref","unstructured":"Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer., 19:209\u2013286, (2010)","DOI":"10.1017\/S0962492910000048"},{"key":"3184_CR23","unstructured":"Hundsdorfer, W., Verwer, J.: Numerical Solution of Time-Dependent Advection-Diffusion Reaction Equations. Springer, (2007)"},{"key":"3184_CR24","doi-asserted-by":"publisher","first-page":"855","DOI":"10.1007\/s11590-023-02069-7","volume":"18","author":"N Karagiannis-Axypolitidis","year":"2024","unstructured":"Karagiannis-Axypolitidis, N., Panebianco, F., Bonanomi, G., Giannino, F.: Plants\u2019 competition under autotoxicity effect: an evolutionary game. Optim. Lett. 18, 855\u2013872 (2024)","journal-title":"Optim. Lett."},{"key":"3184_CR25","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109229","volume":"407","author":"H Lu","year":"2020","unstructured":"Lu, H., Tartakovsky, D.M.: Lagrangian dynamic mode decomposition for construction of reduced-order models of advection-dominated phenomena. J. Comput. Phys. 407, 109229 (2020)","journal-title":"J. Comput. Phys."},{"key":"3184_CR26","doi-asserted-by":"publisher","first-page":"156","DOI":"10.1016\/j.apnum.2024.08.023","volume":"207","author":"G Pagano","year":"2025","unstructured":"Pagano, G.: Stabilized explicit peer methods with parallelism across the stages for stiff problems. Appl. Numer. Math. 207, 156\u2013173 (2025)","journal-title":"Appl. Numer. Math."},{"key":"3184_CR27","doi-asserted-by":"crossref","unstructured":"Permyakova, E. V., Goldobin, D. S.: High-order schemes of exponential time differencing for stiff systems with nondiagonal linear part. J. Comput. Phys., 510:113493, (2025)","DOI":"10.1016\/j.jcp.2024.113493"},{"key":"3184_CR28","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1093\/comjnl\/5.4.329","volume":"5","author":"HH Rosenbrock","year":"1963","unstructured":"Rosenbrock, H.H.: Some general implicit processes for the numerical solution of differential equations. Comput. J. 5, 329\u2013330 (1963)","journal-title":"Comput. J."},{"key":"3184_CR29","doi-asserted-by":"crossref","unstructured":"Van Daele, M., Vanden Berghe, G., De Meyer, H.: Explicit fifth order Runge-Kutta methods with five stages for quadratic ODEs. Numerical Analysis and Its Applications, pages 528\u2013535, (1997)","DOI":"10.1007\/3-540-62598-4_134"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03184-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-026-03184-0","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03184-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T08:22:49Z","timestamp":1770970969000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-026-03184-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,23]]},"references-count":29,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,2]]}},"alternative-id":["3184"],"URL":"https:\/\/doi.org\/10.1007\/s10915-026-03184-0","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,23]]},"assertion":[{"value":"28 March 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 November 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 January 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 January 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"57"}}