{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T00:46:47Z","timestamp":1767833207034,"version":"3.49.0"},"reference-count":50,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,4,21]],"date-time":"2022-04-21T00:00:00Z","timestamp":1650499200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In recent years, the efficient numerical solution of Hamiltonian problems has led to the definition of a class of energy-conserving Runge\u2013Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Such methods admit an interesting interpretation in terms of continuous-stage Runge\u2013Kutta methods. In this review paper, we recall this aspect and extend it to higher-order differential problems.<\/jats:p>","DOI":"10.3390\/axioms11050192","type":"journal-article","created":{"date-parts":[[2022,4,22]],"date-time":"2022-04-22T05:07:22Z","timestamp":1650604042000},"page":"192","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Continuous-Stage Runge\u2013Kutta Approximation to Differential Problems"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3372-0162","authenticated-orcid":false,"given":"Pierluigi","family":"Amodio","sequence":"first","affiliation":[{"name":"Dipartimento di Matematica, Universit\u00e0 di Bari, Via Orabona 4, 70125 Bari, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6290-4107","authenticated-orcid":false,"given":"Luigi","family":"Brugnano","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica e Informatica \u201cU. Dini\u201d, Universit\u00e0 di Firenze, Viale Morgagni 67\/A, 50134 Firenze, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9716-7370","authenticated-orcid":false,"given":"Felice","family":"Iavernaro","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica, Universit\u00e0 di Bari, Via Orabona 4, 70125 Bari, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1090\/S0025-5718-1972-0305608-0","article-title":"An algebraic theory of integration methods","volume":"26","author":"Butcher","year":"1972","journal-title":"Math. Comp."},{"key":"ref_2","unstructured":"Butcher, J.C. (1987). The Numerical Analysis of Ordinary Differential Equations: Runge\u2013Kutta and General Linear Methods, John Wiley & Sons."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/S0168-9274(96)00048-7","article-title":"Runge\u2013Kutta methods: Some historical notes","volume":"22","author":"Butcher","year":"1996","journal-title":"Appl. Numer. Math."},{"key":"ref_4","first-page":"17","article-title":"Hamiltonian Boundary Value Methods (Energy Preserving Discrete Line Integral Methods)","volume":"5","author":"Brugnano","year":"2010","journal-title":"JNAIAM J. Numer. Anal. Ind. Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"8475","DOI":"10.1016\/j.amc.2012.01.074","article-title":"A simple framework for the derivation and analysis of effective one-step methods for ODEs","volume":"218","author":"Brugnano","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_6","first-page":"73","article-title":"Energy-preserving variant of collocation methods","volume":"5","author":"Hairer","year":"2010","journal-title":"JNAIAM J. Numer. Anal. Ind. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1379","DOI":"10.1007\/s11075-019-00655-4","article-title":"Energy-preserving trigonometrically fitted continuous stage Runge\u2013Kutta-Nystr\u00f6m methods for oscillatory Hamiltonian systems","volume":"81","author":"Li","year":"2019","journal-title":"Numer. Algorithms"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1007\/s10543-014-0474-4","article-title":"An energy-preserving exponentially-fitted continuous stage Runge\u2013Kutta method for Hamiltonian systems","volume":"54","author":"Miyatake","year":"2014","journal-title":"BIT"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1993","DOI":"10.1137\/15M1020861","article-title":"A characterization of energy preserving methods and the construction of parallel integrators for Hamiltonian systems","volume":"54","author":"Miyatake","year":"2016","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"045206","DOI":"10.1088\/1751-8113\/41\/4\/045206","article-title":"A new class of energy-preserving numerical integration methods","volume":"41","author":"Quispel","year":"2008","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1071","DOI":"10.1007\/s10483-007-0809-y","article-title":"Continuous finite element methods for Hamiltonian systems","volume":"28","author":"Tang","year":"2007","journal-title":"Appl. Math. Mech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/j.amc.2018.07.044","article-title":"A note on continuous-stage Runge\u2013Kutta methods","volume":"339","author":"Tang","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.amc.2016.04.026","article-title":"Construction of symplectic (partitioned) Runge\u2013Kutta methods with continuous stage","volume":"286","author":"Tang","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2158","DOI":"10.1016\/j.amc.2012.08.062","article-title":"Time finite element methods: A unified framework for numerical discretizations of ODEs","volume":"219","author":"Tang","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/j.amc.2014.02.042","article-title":"Construction of Runge\u2013Kutta type methods for solving ordinary differential equations","volume":"234","author":"Tang","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.amc.2019.05.013","article-title":"Symmetric integrators based on continuous-stage Runge\u2013Kutta-Nystr\u00f6m methods for reversible systems","volume":"361","author":"Tang","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_17","first-page":"1","article-title":"Continuous stage stochastic Runge\u2013Kutta methods","volume":"61","author":"Xin","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"814","DOI":"10.4208\/nmtma.OA-2019-0115","article-title":"A continuous-stage modified Leap-frog scheme for high-dimensional semi-linear Hamiltonian wave equations","volume":"13","author":"Wang","year":"2020","journal-title":"Numer. Math. Theory Methods Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1515\/spma-2021-0101","article-title":"On eigenvalues of a matrix arising in energy-preserving\/dissipative continuous-stage Runge\u2013Kutta methods","volume":"10","author":"Yamamoto","year":"2022","journal-title":"Spec. Matrices"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"881","DOI":"10.1007\/s10444-014-9390-z","article-title":"Energy-conserving methods for Hamiltonian Boundary Value Problems and applications in astrodynamics","volume":"41","author":"Amodio","year":"2015","journal-title":"Adv. Comput. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"124634","DOI":"10.1016\/j.amc.2019.124634","article-title":"A note on the continuous-stage Runge\u2013Kutta-(Nystr\u00f6m) formulation of Hamiltonian Boundary Value Methods (HBVMs)","volume":"363","author":"Amodio","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1489","DOI":"10.1007\/s11075-019-00733-7","article-title":"Analysis of Spectral Hamiltonian Boundary Value Methods (SHBVMs) for the numerical solution of ODE problems","volume":"83","author":"Amodio","year":"2020","journal-title":"Numer. Algorithms"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Amodio, P., Brugnano, L., and Iavernaro, F. (2022). Arbitrarily high-order energy-conserving methods for Poisson problems. Numer. Algorithms.","DOI":"10.1007\/s11075-022-01285-z"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"112918","DOI":"10.1016\/j.cam.2020.112918","article-title":"Spectrally accurate space-time solution of Manakov systems","volume":"377","author":"Barletti","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Brugnano, L., Frasca-Caccia, G., and Iavernaro, F. (2019). Line Integral Solution of Hamiltonian PDEs. Mathematics, 7.","DOI":"10.3390\/math7030275"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.apnum.2017.12.014","article-title":"Line integral solution of Hamiltonian systems with holonomic constraints","volume":"127","author":"Brugnano","year":"2018","journal-title":"Appl. Numer. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.cam.2018.10.014","article-title":"Energy-conserving Hamiltonian Boundary Value Methods for the numerical solution of the Korteweg-de Vries equation","volume":"351","author":"Brugnano","year":"2019","journal-title":"J. Comput. Appl. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1343","DOI":"10.1002\/num.22353","article-title":"Spectrally accurate energy-preserving methods for the numerical solution of the \u201cGood\u201d Boussinesq equation","volume":"35","author":"Brugnano","year":"2019","journal-title":"Numer. Methods Partial. Differ. Equ."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Brugnano, L., and Iavernaro, F. (2016). Line Integral Methods for Conservative Problems, Chapman & Hall\/CRC. Available online: http:\/\/web.math.unifi.it\/users\/brugnano\/LIMbook\/.","DOI":"10.1201\/b19319"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Brugnano, L., and Iavernaro, F. (2018). Line Integral Solution of Differential Problems. Axioms, 7.","DOI":"10.3390\/axioms7020036"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.1007\/s11075-018-0586-z","article-title":"Spectrally accurate space-time solution of Hamiltonian PDEs","volume":"81","author":"Brugnano","year":"2019","journal-title":"Numer. Algorithms"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1016\/j.cam.2011.07.022","article-title":"A note on the efficient implementation of Hamiltonian BVMs","volume":"236","author":"Brugnano","year":"2011","journal-title":"J. Comput. Appl. Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1860","DOI":"10.1016\/j.cpc.2012.04.002","article-title":"A two-step, fourth-order method with energy preserving properties","volume":"183","author":"Brugnano","year":"2012","journal-title":"Comput. Phys. Commun."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"112994","DOI":"10.1016\/j.cam.2020.112994","article-title":"Arbitrarily high-order energy-preserving methods for simulating the gyrocenter dynamics of charged particles","volume":"380","author":"Brugnano","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1007\/s11075-018-0552-9","article-title":"On the effectiveness of spectral methods for the numerical solution of multi-frequency highly-oscillatory Hamiltonian problems","volume":"81","author":"Brugnano","year":"2019","journal-title":"Numer. Algorithms"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/j.jcp.2019.06.068","article-title":"High-order energy-conserving Line Integral Methods for charged particle dynamics","volume":"396","author":"Brugnano","year":"2019","journal-title":"J. Comput. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1007\/s11075-013-9769-9","article-title":"Multiple invariants conserving Runge\u2013Kutta type methods for Hamiltonian problems","volume":"65","author":"Brugnano","year":"2014","journal-title":"Numer. Algorithms"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"670","DOI":"10.1016\/j.amc.2019.06.031","article-title":"High order symplectic integrators based on continuous-stage Runge\u2013Kutta-Nystr\u00f6m methods","volume":"361","author":"Tang","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.amc.2017.11.054","article-title":"Symplecticity-preserving continuous stage Runge\u2013Kutta-Nystr\u00f6m methods","volume":"323","author":"Tang","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_40","unstructured":"Hairer, E., and Wanner, G. (2002). Solving Ordinary Differential Equations II, Springer. [2nd ed.]."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Dahlquist, G., and Bj\u00f6rk, \u00c5. (2008). Numerical Methods in Scientific Computing, SIAM.","DOI":"10.1137\/1.9780898717785"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Blanes, S., and Casas, F. (2016). A Concise Introduction to Geometric Numerical Integration, CRC Press.","DOI":"10.1201\/b21563"},{"key":"ref_43","unstructured":"Hairer, E., Lubich, C., and Wanner, G. (2006). Geometric Numerical Integration, Springer."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Leimkuhler, B., and Reich, S. (2004). Simulating Hamiltonian Dynamics, Cambridge University Press.","DOI":"10.1017\/CBO9780511614118"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Sanz-Serna, J.M., and Calvo, M.P. (1994). Numerical Hamiltonian Problems, Chapman & Hall.","DOI":"10.1007\/978-1-4899-3093-4"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/s00211-015-0766-x","article-title":"Long-term analysis of the St\u00f6rmer-Verlet method for Hamiltonian systems with a solution-dependent high frequency","volume":"134","author":"Hairer","year":"2016","journal-title":"Numer. Math."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"987","DOI":"10.4208\/cicp.OA-2021-0154","article-title":"Tuning symplectic integrators is easy and worthwhile","volume":"31","author":"McLachlan","year":"2022","journal-title":"Commun. Comput. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1137\/151002769","article-title":"Symplectic Runge\u2013Kutta schemes for adjoint equations, automatic differentiation, optimal control, and more","volume":"58","year":"2016","journal-title":"SIAM Rev."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"977","DOI":"10.1007\/s10543-021-00846-3","article-title":"A long-term numerical energy-preserving analysis of symmetric and\/or symplectic extended RKN integrators for efficiently solving highly oscillatory Hamiltonian systems","volume":"61","author":"Wang","year":"2021","journal-title":"Bit Numer. Math."},{"key":"ref_50","unstructured":"(2022, April 15). Available online: https:\/\/www.mrsir.it\/en\/about-us\/."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/5\/192\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:57:58Z","timestamp":1760137078000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/5\/192"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,21]]},"references-count":50,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["axioms11050192"],"URL":"https:\/\/doi.org\/10.3390\/axioms11050192","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,21]]}}}