{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T08:07:57Z","timestamp":1784275677415,"version":"3.55.0"},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,5,25]],"date-time":"2026-05-25T00:00:00Z","timestamp":1779667200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,5,25]],"date-time":"2026-05-25T00:00:00Z","timestamp":1779667200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100017142","name":"Gruppo Nazionale per il Calcolo Scientifico","doi-asserted-by":"publisher","award":["GNCS-INdAM project"],"award-info":[{"award-number":["GNCS-INdAM project"]}],"id":[{"id":"10.13039\/100017142","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003407","name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["PRIN PNRR 2022 project P20228C2PP (CUP: F53D23010020001) \u201cBAT-MEN\u201d"],"award-info":[{"award-number":["PRIN PNRR 2022 project P20228C2PP (CUP: F53D23010020001) \u201cBAT-MEN\u201d"]}],"id":[{"id":"10.13039\/501100003407","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003407","name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["PRIN PNRR 2022 project P2022WC2ZZ (CUP: E53D23018030001) \u201cMatForPat\u201d"],"award-info":[{"award-number":["PRIN PNRR 2022 project P2022WC2ZZ (CUP: E53D23018030001) \u201cMatForPat\u201d"]}],"id":[{"id":"10.13039\/501100003407","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004837","name":"Ministerio de Ciencia e Innovaci\u00f3n","doi-asserted-by":"publisher","award":["Project PID2022\u2013141385NB\u2013I00"],"award-info":[{"award-number":["Project 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":["Adv Comput Math"],"published-print":{"date-parts":[[2026,6]]},"abstract":"Abstract<\/jats:title>\n \n This paper addresses the efficient numerical solution of large and stiff initial value problems (IVPs) arising from the space discretization of systems of nonlinear advection\u2013diffusion-reaction partial differential equations (PDEs). To this end, we introduce a new family of linearly implicit two-step peer methods that leverage specialized preconditioners and exploit the reuse of previously computed stages. The proposed methods are constructed to ensure strong stability properties, specifically, L-stability or L(\n \n \n $$\\theta $$<\/jats:tex-math>\n \n \u03b8<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n )-stability with\n \n \n $$\\theta $$<\/jats:tex-math>\n \n \u03b8<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n approaching 90\n \n \n $$^{\\circ }$$<\/jats:tex-math>\n \n \n \n \n \u2218<\/mml:mo>\n <\/mml:mmultiscripts>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , while maintaining low error constants. Compared to recently developed linearly implicit peer schemes, the new methods significantly reduce both the number of required function evaluations and linear systems to solve at each time step, also resulting in notably diminished error constants. Numerical experiments on nonlinear advection\u2013diffusion-reaction problems testify to the efficiency of the new methods and confirm their accuracy and stability properties.\n <\/jats:p>","DOI":"10.1007\/s10444-026-10315-6","type":"journal-article","created":{"date-parts":[[2026,5,25]],"date-time":"2026-05-25T14:53:02Z","timestamp":1779720782000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["L($$\\theta $$)-stable peer methods with reused stages for advection\u2013diffusion-reaction problems"],"prefix":"10.1007","volume":"52","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6373-9473","authenticated-orcid":false,"given":"Giovanni","family":"Pagano","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luis","family":"R\u00e1ndez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,25]]},"reference":[{"key":"10315_CR1","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1016\/j.apnum.2023.01.015","volume":"186","author":"A Abdi","year":"2023","unstructured":"Abdi, A., Hojjati, G., Jackiewicz, Z., Podhaisky, H., Sharifi, M.: On the implementation of explicit two-step peer methods with Runge-Kutta stability. Appl. Numer. Math. 186, 213\u2013227 (2023)","journal-title":"Appl. Numer. Math."},{"key":"10315_CR2","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1016\/j.apnum.2023.04.001","volume":"200","author":"L Aceto","year":"2024","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)","journal-title":"Appl. Numer. Math."},{"issue":"3","key":"10315_CR3","doi-asserted-by":"publisher","first-page":"A1652","DOI":"10.1137\/24M1667336","volume":"47","author":"L Aceto","year":"2025","unstructured":"Aceto, L., Conte, D., Pagano, G.: Modified tase Runge-Kutta methods for integrating stiff differential equations. SIAM J. Sci. Comput. 47(3), A1652\u2013A1680 (2025)","journal-title":"SIAM J. Sci. Comput."},{"key":"10315_CR4","doi-asserted-by":"crossref","unstructured":"Bassenne, M., Fu, L., Mani, A.: Time\u2013accurate and highly\u2013stable explicit operators for stiff differential equations. J. Comput. Phys., 424 (2021)","DOI":"10.1016\/j.jcp.2020.109847"},{"key":"10315_CR5","doi-asserted-by":"crossref","unstructured":"Calvo, M., Fu, L., Montijano, J.\u00a0I., 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":"10315_CR6","doi-asserted-by":"crossref","unstructured":"Calvo, M., Montijano, J.\u00a0I., R\u00e1ndez, L., Saenz de-la Torre,\u00a0A.: Explicit two\u2013step peer methods with reused stages. Appl. Numer. Math. 195, 75\u201388 (2024)","DOI":"10.1016\/j.apnum.2023.09.008"},{"issue":"4","key":"10315_CR7","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1016\/j.apnum.2010.11.004","volume":"61","author":"M Calvo","year":"2011","unstructured":"Calvo, M., Montijano, J.I., R\u00e1ndez, L., Van Daele, M.: On the derivation of explicit two-step peer methods. Appl. Numer. Math. 61(4), 395\u2013409 (2011)","journal-title":"Appl. Numer. Math."},{"key":"10315_CR8","doi-asserted-by":"crossref","unstructured":"Calvo, M., Montijano, J.\u00a0I., 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":"10315_CR9","doi-asserted-by":"crossref","unstructured":"Calvo, M., Montijano, J.\u00a0I., R\u00e1ndez, L.: Modified singly\u2013Runge\u2013Kutta\u2013TASE methods for the numerical solution of stiff differential equations. J. Sci. Comput. 103(1)(3) (2025)","DOI":"10.1007\/s10915-025-02813-4"},{"issue":"2","key":"10315_CR10","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."},{"issue":"6","key":"10315_CR11","doi-asserted-by":"publisher","first-page":"A3638","DOI":"10.1137\/24M1631869","volume":"46","author":"D Conte","year":"2024","unstructured":"Conte, D., Martin-Vaquero, J., Pagano, G., Paternoster, B.: Stability theory of TASE-Runge-Kutta methods with inexact Jacobian. SIAM J. Sci. Comput. 46(6), A3638\u2013A3657 (2024)","journal-title":"SIAM J. Sci. Comput."},{"key":"10315_CR12","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":"10315_CR13","doi-asserted-by":"crossref","unstructured":"Conte, D., Pagano, G., Paternoster, B.: Time\u2013accurate and highly\u2013stable explicit peer methods for stiff differential problems. Comm. Nonlinear Sci. Numer. Simul. 119 (2023)","DOI":"10.1016\/j.cnsns.2023.107136"},{"issue":"2","key":"10315_CR14","doi-asserted-by":"publisher","first-page":"1294","DOI":"10.1093\/imanum\/dry087","volume":"40","author":"G Frasca-Caccia","year":"2020","unstructured":"Frasca-Caccia, G., Hydon, P.E.: Simple bespoke preservation of two conservation laws. IMA J. Numer. Anal. 40(2), 1294\u20131329 (2020)","journal-title":"IMA J. Numer. Anal."},{"key":"10315_CR15","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"},{"issue":"2","key":"10315_CR16","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":"10315_CR17","volume-title":"The theory of matrices","author":"FR Gantmacher","year":"1998","unstructured":"Gantmacher, F.R.: The theory of matrices. AMS Chelsea Publishing, Providence, RI (1998)"},{"key":"10315_CR18","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":"10315_CR19","doi-asserted-by":"crossref","unstructured":"Hairer, E., Wanner, G.: Solving ordinary differential equations II (stiff and differential\u2013algebraic problems). Springer Series in Computational Mathematics (14). Springer, Berlin (1996)","DOI":"10.1007\/978-3-642-05221-7"},{"key":"10315_CR20","unstructured":"Hairer, E., Wanner, G., N\u00f8rsett, S.\u00a0P.: Solving ordinary differential equations I (nonstiff problems). Springer Series in Computational Mathematics (8). Springer Berlin (1993)"},{"key":"10315_CR21","doi-asserted-by":"publisher","first-page":"776","DOI":"10.1016\/j.cam.2015.11.005","volume":"296","author":"Z Horv\u00e1th","year":"2016","unstructured":"Horv\u00e1th, Z., Podhaisky, H., Weiner, R.: Strong stability preserving explicit peer methods. J. Comput. Appl. Math. 296, 776\u2013788 (2016)","journal-title":"J. Comput. Appl. Math."},{"key":"10315_CR22","doi-asserted-by":"publisher","DOI":"10.1002\/9780470522165","volume-title":"General linear methods for ordinary differential equations","author":"Z Jackiewicz","year":"2009","unstructured":"Jackiewicz, Z.: General linear methods for ordinary differential equations. John Wiley & Sons Inc, Hoboken, NJ (2009)"},{"key":"10315_CR23","doi-asserted-by":"crossref","unstructured":"Klinge, M., Hern\u00e1ndez-Abreu, D., Weiner, R.: A comparison of one\u2013step and two\u2013step W\u2013methods and peer methods with approximate matrix factorization. J. Comput. Appl. Math. 387 (2021)","DOI":"10.1016\/j.cam.2019.112519"},{"key":"10315_CR24","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1007\/s10543-017-0691-8","volume":"58","author":"M Klinge","year":"2018","unstructured":"Klinge, M., Weiner, R., Podhaisky, H.: Optimally zero stable explicit peer methods with variable nodes. BIT Numer. Math. 58, 331\u2013345 (2018)","journal-title":"BIT Numer. Math."},{"issue":"2","key":"10315_CR25","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1007\/s10543-018-0734-9","volume":"59","author":"JI Montijano","year":"2019","unstructured":"Montijano, J.I., Podhaisky, H., R\u00e1ndez, L., Calvo, M.: A family of L-stable singly implicit peer methods for solving stiff IVPs. BIT Num. Math. 59(2), 483\u2013502 (2019)","journal-title":"BIT Num. Math."},{"key":"10315_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."},{"issue":"1","key":"10315_CR27","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1137\/S0036142902411057","volume":"42","author":"BA Schmitt","year":"2004","unstructured":"Schmitt, B.A., Weiner, R.: Parallel two-step W-methods with peer variables. SIAM J. Numer. Anal. 42(1), 265\u2013282 (2004)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"10315_CR28","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1007\/s10543-005-2635-y","volume":"45","author":"BA Schmitt","year":"2005","unstructured":"Schmitt, B.A., Weiner, R., Podhaisky, H.: Multi-implicit peer two-step $$W$$-methods for parallel time integration. BIT 45(1), 197\u2013217 (2005)","journal-title":"BIT"},{"issue":"4","key":"10315_CR29","doi-asserted-by":"publisher","first-page":"609","DOI":"10.1016\/j.camwa.2007.04.026","volume":"55","author":"R Weiner","year":"2008","unstructured":"Weiner, R., Biermann, K., Schmitt, B.A., Podhaisky, H.: Explicit two-step peer methods. Comput. Math. Appl. 55(4), 609\u2013619 (2008)","journal-title":"Comput. Math. Appl."},{"issue":"3\u20134","key":"10315_CR30","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1016\/j.apnum.2003.10.005","volume":"48","author":"R Weiner","year":"2004","unstructured":"Weiner, R., Schmitt, B.A., Podhaisky, H.: Parallel \u2018Peer\u2019 two-step $$W$$-methods and their application to MOL-systems. Appl. Numer. Math. 48(3\u20134), 425\u2013439 (2004)","journal-title":"Appl. Numer. Math."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-026-10315-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-026-10315-6","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-026-10315-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T07:54:02Z","timestamp":1784274842000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-026-10315-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,25]]},"references-count":30,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,6]]}},"alternative-id":["10315"],"URL":"https:\/\/doi.org\/10.1007\/s10444-026-10315-6","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,25]]},"assertion":[{"value":"1 August 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 May 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 May 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","label":"Conflict of interest","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"41"}}