DOI: 10.1017/s0962492900002282. Symplectic integrators for Hamiltonian problems: an overview

Symplectic integrators for Hamiltonian problems: an overview

10.1017/s0962492900002282

Crossref journal-article

Cambridge University Press (CUP)

Acta Numerica (56)

Abstract

In the sciences, situations where dissipation is not significant may invariably be modelled by Hamiltonian systems of ordinary, or partial, differential equations. Symplectic integrators are numerical methods specifically aimed at advancing in time the solution of Hamiltonian systems. Roughly speaking, ‘symplecticness’ is a characteristic property possessed by the solutions of Hamiltonian problems. A numerical method is called symplectic if, when applied to Hamiltonian problems, it generates numerical solutions which inherit the property of symplecticness.

Bibliography

Sanz-Serna, J. M. (1992). Symplectic integrators for Hamiltonian problems: an overview. Acta Numerica, 1, 243â286.

Authors 1

J. M. Sanz-Serna (first)

References 72 Referenced 250

10.1016/0021-9991(74)90011-4
{'key': 'S0962492900002282_ref069', 'first-page': '78', 'article-title': 'Hamiltonian Runge–Kutta type methods and their variational formulation', 'volume': '2', 'author': 'Suris', 'year': '1990', 'journal-title': 'Math. Sim.'} / Math. Sim. / Hamiltonian Runge–Kutta type methods and their variational formulation by Suris (1990)
{'key': 'S0962492900002282_ref066', 'first-page': '187', 'volume-title': 'Numerical Analysis', 'author': 'Sanz-Serna', 'year': '1986'} / Numerical Analysis by Sanz-Serna (1986)
10.1093/imanum/11.4.449
{'key': 'S0962492900002282_ref063', 'volume-title': 'Applied Mathematics and Computation Reports', 'author': 'Sanz-Serna', 'year': '1991'} / Applied Mathematics and Computation Reports by Sanz-Serna (1991)
10.1093/oso/9780198534389.003.0004 / Advances in Numerical Analysis by Sanz-Serna (1991)
{'key': 'S0962492900002282_ref060', 'volume-title': 'Proc. 1989 London Numerical ODE Conference', 'author': 'Sanz-Serna', 'year': '1989'} / Proc. 1989 London Numerical ODE Conference by Sanz-Serna (1989)
10.1016/0898-1221(90)90357-P
{'key': 'S0962492900002282_ref056', 'first-page': '272', 'article-title': 'Leap-frog schemes of two kinds of Hamiltonian systems for wave equations', 'volume': '10', 'author': 'Meng-zhao', 'year': '1988', 'journal-title': 'Math. Num. Sin.'} / Math. Num. Sin. / Leap-frog schemes of two kinds of Hamiltonian systems for wave equations by Meng-zhao (1988)
{'key': 'S0962492900002282_ref053', 'first-page': '90', 'volume-title': 'Working Document', 'author': 'Okunbor', 'year': '1990'} / Working Document by Okunbor (1990)
10.1016/0021-8928(84)90078-9
{'key': 'S0962492900002282_ref050', 'volume-title': 'Stable and Random Motions in Dynamical Systems', 'author': 'Moser', 'year': '1973'} / Stable and Random Motions in Dynamical Systems by Moser (1973)
10.1016/0167-2789(84)90438-X
{'key': 'S0962492900002282_ref046', 'volume-title': 'Hamiltonian Dynamical Systems', 'author': 'MacKay', 'year': '1987'} / Hamiltonian Dynamical Systems by MacKay (1987)
{'key': 'S0962492900002282_ref045', 'volume-title': 'Proceedings of the ‘Dynamics of Numerics and Numerics of Dynamics’ Conference', 'author': 'MacKay', 'year': '1991'} / Proceedings of the ‘Dynamics of Numerics and Numerics of Dynamics’ Conference by MacKay (1991)
{'key': 'S0962492900002282_ref044', 'first-page': '164', 'article-title': 'A symplectic difference scheme for the infinite dimensional Hamiltonian system', 'volume': '6', 'author': 'Chun-wang', 'year': '1988', 'journal-title': 'J. Comput. Math.'} / J. Comput. Math. / A symplectic difference scheme for the infinite dimensional Hamiltonian system by Chun-wang (1988)
10.1137/0716004
{'key': 'S0962492900002282_ref047', 'volume-title': 'Program in Applied Mathematics', 'author': 'McLachlan', 'year': '1991'} / Program in Applied Mathematics by McLachlan (1991)
{'key': 'S0962492900002282_ref027', 'first-page': '1', 'volume-title': "Proceedings of the 1st Chinese Conference for Numerical Methods for PDE's, March 1986, Shanghai", 'author': 'Kang', 'year': '1987'} / Proceedings of the 1st Chinese Conference for Numerical Methods for PDE's, March 1986, Shanghai by Kang (1987)
10.1109/TNS.1983.4332919
10.1016/0898-1221(88)90139-3 / Comput. Math. Appl. / The generating function for the solution of ODE's and its discrete methods by Yuhua (1988)
{'key': 'S0962492900002282_ref061', 'volume-title': 'Applied Mathematics and Computation Reports', 'author': 'Sanz-Serna', 'year': '1990'} / Applied Mathematics and Computation Reports by Sanz-Serna (1990)
{'key': 'S0962492900002282_ref049', 'volume-title': 'Schwerpunktprogramm der Deutschen Forschungsgemeinschaft, Anwendungsbezogene Optimierung und Steuerung', 'author': 'Miesbach', 'year': '1990'} / Schwerpunktprogramm der Deutschen Forschungsgemeinschaft, Anwendungsbezogene Optimierung und Steuerung by Miesbach (1990)
10.1007/978-3-662-06793-2
10.1098/rspa.1991.0027
10.1016/0021-9991(91)90299-Z
10.1016/0375-9601(90)90092-3
10.1007/978-1-4612-1140-2
Lasagni F. (1990), ‘Integration methods for Hamiltonian differential equations’, unpublished manuscript.
10.1137/0147005
{'key': 'S0962492900002282_ref029', 'first-page': '71', 'article-title': 'Construction of canonical difference schemes for Hamiltonian formalism via generating functions', 'volume': '7', 'author': 'Kang', 'year': '1989', 'journal-title': 'J. Comput. Math.'} / J. Comput. Math. / Construction of canonical difference schemes for Hamiltonian formalism via generating functions by Kang (1989)
{'key': 'S0962492900002282_ref031', 'volume-title': 'Applied Mathematics and Computation Reports', 'author': 'de Frutos', 'year': '1991'} / Applied Mathematics and Computation Reports by de Frutos (1991)
10.1007/BF01954907
10.1137/0907067
Neri F. (1987), ‘Lie algebras and canonical integration’, University of Maryland Technical Report.
10.1093/imanum/10.1.1
{'key': 'S0962492900002282_ref039', 'volume-title': 'Numerical Analysis Report DAMTP', 'author': 'Iserles', 'year': '1990'} / Numerical Analysis Report DAMTP by Iserles (1990)
{'key': 'S0962492900002282_ref068', 'first-page': '202', 'article-title': 'Canonical transformations generated by methods of Runge-Kutta type for the numerical integration of the system x″ = −∂U∂x', 'volume': '29', 'author': 'Suris', 'year': '1989', 'journal-title': 'Zh. Vychisl. Mat. i Mat. Fiz.'} / Zh. Vychisl. Mat. i Mat. Fiz. / Canonical transformations generated by methods of Runge-Kutta type for the numerical integration of the system x″ = −∂U∂x by Suris (1989)
10.1016/0375-9601(88)90773-6
{'key': 'S0962492900002282_ref012', 'article-title': 'Order conditions for canonical Runge–Kutta–Nyström methods', 'author': 'Calvo', 'year': '1991', 'journal-title': 'BIT'} / BIT / Order conditions for canonical Runge–Kutta–Nyström methods by Calvo (1991)
Channell P. J. , ‘Symplectic integration algorithms’, Los Alamos National Laboratory Report, Report AT-6ATN 83–9.
{'key': 'S0962492900002282_ref028', 'first-page': '371', 'article-title': 'Symplectic difference schemes for linear Hamiltonian canonical systems', 'volume': '8', 'author': 'Kang', 'year': '1990', 'journal-title': 'J. Comput. Math.'} / J. Comput. Math. / Symplectic difference schemes for linear Hamiltonian canonical systems by Kang (1990)
10.1007/BF01932265
10.1093/oso/9780198534389.003.0005 / Advances in Numerical Analysis by Beyn (1991)
10.1007/978-3-662-02535-2
10.1007/978-3-662-12607-3
{'key': 'S0962492900002282_ref021', 'volume-title': 'Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations', 'author': 'Dekker', 'year': '1984'} / Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations by Dekker (1984)
10.1137/0728058
10.1093/imanum/7.2.235
{'key': 'S0962492900002282_ref024', 'first-page': '42', 'volume-title': 'Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations', 'author': 'Kang', 'year': '1985'} / Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations by Kang (1985)
10.1016/0045-7825(90)90046-O
10.1007/BF01397651
Calvo M.P. (1991), Ph.D. Thesis, University of Valladolid (to appear).
{'key': 'S0962492900002282_ref026', 'first-page': '1', 'volume-title': 'Tenth International Conference on Numerical Methods in Fluid Dynamics', 'author': 'Kang', 'year': '1986'} / Tenth International Conference on Numerical Methods in Fluid Dynamics by Kang (1986)
{'key': 'S0962492900002282_ref001', 'volume-title': 'Applied Mathematics and Computation Reports', 'author': 'Abia', 'year': '1990'} / Applied Mathematics and Computation Reports by Abia (1990)
10.1016/0021-9991(85)90180-9
10.1007/978-3-642-61551-1
10.1007/978-1-4757-2063-1
{'key': 'S0962492900002282_ref054', 'first-page': '91', 'volume-title': 'Working Document', 'author': 'Okunbor', 'year': '1991'} / Working Document by Okunbor (1991)
{'key': 'S0962492900002282_ref010', 'volume-title': 'The Numerical Analysis of Ordinary Differential Equations', 'author': 'Butcher', 'year': '1987'} / The Numerical Analysis of Ordinary Differential Equations by Butcher (1987)
{'key': 'S0962492900002282_ref014', 'volume-title': 'Applied Mathematics and Computation Reports', 'author': 'Calvo', 'year': '1991'} / Applied Mathematics and Computation Reports by Calvo (1991)
10.1088/0951-7715/3/2/001
{'key': 'S0962492900002282_ref013', 'volume-title': 'Applied Mathematics and Computation Reports', 'author': 'Calvo', 'year': '1991'} / Applied Mathematics and Computation Reports by Calvo (1991)
10.1093/imanum/7.1.1
10.1007/BF02238800
{'key': 'S0962492900002282_ref023', 'volume-title': 'Applied Mathematics and Computation Reports', 'author': 'Eirola', 'year': '1990'} / Applied Mathematics and Computation Reports by Eirola (1990)
{'key': 'S0962492900002282_ref025', 'first-page': '279', 'article-title': 'Difference schemes for Hamiltonian formalism and symplectic geometry', 'volume': '4', 'author': 'Kang', 'year': '1986', 'journal-title': 'J. Comput. Math.'} / J. Comput. Math. / Difference schemes for Hamiltonian formalism and symplectic geometry by Kang (1986)
10.1016/0167-2789(90)90019-L
10.1007/BF01385623
10.1007/978-1-4899-3071-2
{'key': 'S0962492900002282_ref041', 'volume-title': 'Vorlesungen Über die Entwicklung der Mathematik im 19.Jahrhundert', 'author': 'Klein', 'year': '1926'} / Vorlesungen Über die Entwicklung der Mathematik im 19.Jahrhundert by Klein (1926)
10.1007/BF00945133

Dates

Type	When
Created	16 years, 9 months ago (Nov. 7, 2008, 11:31 a.m.)
Deposited	1 year, 5 months ago (March 2, 2024, 4:52 a.m.)
Indexed	6 days, 11 hours ago (Aug. 20, 2025, 8:34 a.m.)
Issued	33 years, 7 months ago (Jan. 1, 1992)
Published	33 years, 7 months ago (Jan. 1, 1992)
Published Online	16 years, 9 months ago (Nov. 7, 2008)
Published Print	33 years, 7 months ago (Jan. 1, 1992)

Funders 0

None

BibTeX

@article{Sanz_Serna_1992, title={Symplectic integrators for Hamiltonian problems: an overview}, volume={1}, ISSN={1474-0508}, url={http://dx.doi.org/10.1017/s0962492900002282}, DOI={10.1017/s0962492900002282}, journal={Acta Numerica}, publisher={Cambridge University Press (CUP)}, author={Sanz-Serna, J. M.}, year={1992}, month=jan, pages={243–286} }

JSON

{
  "indexed": {
    "date-parts": [
      [
        2025,
        8,
        20
      ]
    ],
    "date-time": "2025-08-20T12:34:56Z",
    "timestamp": 1755693296993
  },
  "reference-count": 72,
  "publisher": "Cambridge University Press (CUP)",
  "license": [
    {
      "start": {
        "date-parts": [
          [
            2008,
            11,
            7
          ]
        ],
        "date-time": "2008-11-07T00:00:00Z",
        "timestamp": 1226016000000
      },
      "content-version": "unspecified",
      "delay-in-days": 6155,
      "URL": "https://www.cambridge.org/core/terms"
    }
  ],
  "content-domain": {
    "domain": [],
    "crossmark-restriction": false
  },
  "published-print": {
    "date-parts": [
      [
        1992,
        1
      ]
    ]
  },
  "abstract": "<jats:p>In the sciences, situations where dissipation is not significant may invariably be modelled by Hamiltonian systems of ordinary, or partial, differential equations. Symplectic integrators are numerical methods specifically aimed at advancing in time the solution of Hamiltonian systems. Roughly speaking, \u2018symplecticness\u2019 is a characteristic property possessed by the solutions of Hamiltonian problems. A numerical method is called symplectic if, when applied to Hamiltonian problems, it generates numerical solutions which inherit the property of symplecticness.</jats:p>",
  "DOI": "10.1017/s0962492900002282",
  "type": "journal-article",
  "created": {
    "date-parts": [
      [
        2008,
        11,
        7
      ]
    ],
    "date-time": "2008-11-07T16:31:48Z",
    "timestamp": 1226075508000
  },
  "page": "243-286",
  "source": "Crossref",
  "is-referenced-by-count": 250,
  "title": "Symplectic integrators for Hamiltonian problems: an overview",
  "prefix": "10.1017",
  "volume": "1",
  "author": [
    {
      "given": "J. M.",
      "family": "Sanz-Serna",
      "sequence": "first",
      "affiliation": []
    }
  ],
  "member": "56",
  "published-online": {
    "date-parts": [
      [
        2008,
        11,
        7
      ]
    ]
  },
  "reference": [
    {
      "key": "S0962492900002282_ref070",
      "doi-asserted-by": "publisher",
      "DOI": "10.1016/0021-9991(74)90011-4"
    },
    {
      "key": "S0962492900002282_ref069",
      "first-page": "78",
      "article-title": "Hamiltonian Runge\u2013Kutta type methods and their variational formulation",
      "volume": "2",
      "author": "Suris",
      "year": "1990",
      "journal-title": "Math. Sim."
    },
    {
      "key": "S0962492900002282_ref066",
      "first-page": "187",
      "volume-title": "Numerical Analysis",
      "author": "Sanz-Serna",
      "year": "1986"
    },
    {
      "key": "S0962492900002282_ref065",
      "doi-asserted-by": "publisher",
      "DOI": "10.1093/imanum/11.4.449"
    },
    {
      "key": "S0962492900002282_ref063",
      "volume-title": "Applied Mathematics and Computation Reports",
      "author": "Sanz-Serna",
      "year": "1991"
    },
    {
      "key": "S0962492900002282_ref062",
      "doi-asserted-by": "crossref",
      "first-page": "147",
      "DOI": "10.1093/oso/9780198534389.003.0004",
      "volume-title": "Advances in Numerical Analysis",
      "volume": "I",
      "author": "Sanz-Serna",
      "year": "1991"
    },
    {
      "key": "S0962492900002282_ref060",
      "volume-title": "Proc. 1989 London Numerical ODE Conference",
      "author": "Sanz-Serna",
      "year": "1989"
    },
    {
      "key": "S0962492900002282_ref057",
      "doi-asserted-by": "publisher",
      "DOI": "10.1016/0898-1221(90)90357-P"
    },
    {
      "key": "S0962492900002282_ref056",
      "first-page": "272",
      "article-title": "Leap-frog schemes of two kinds of Hamiltonian systems for wave equations",
      "volume": "10",
      "author": "Meng-zhao",
      "year": "1988",
      "journal-title": "Math. Num. Sin."
    },
    {
      "key": "S0962492900002282_ref053",
      "first-page": "90",
      "volume-title": "Working Document",
      "author": "Okunbor",
      "year": "1990"
    },
    {
      "key": "S0962492900002282_ref051",
      "doi-asserted-by": "publisher",
      "DOI": "10.1016/0021-8928(84)90078-9"
    },
    {
      "key": "S0962492900002282_ref050",
      "volume-title": "Stable and Random Motions in Dynamical Systems",
      "author": "Moser",
      "year": "1973"
    },
    {
      "key": "S0962492900002282_ref048",
      "doi-asserted-by": "publisher",
      "DOI": "10.1016/0167-2789(84)90438-X"
    },
    {
      "key": "S0962492900002282_ref046",
      "volume-title": "Hamiltonian Dynamical Systems",
      "author": "MacKay",
      "year": "1987"
    },
    {
      "key": "S0962492900002282_ref045",
      "volume-title": "Proceedings of the \u2018Dynamics of Numerics and Numerics of Dynamics\u2019 Conference",
      "author": "MacKay",
      "year": "1991"
    },
    {
      "key": "S0962492900002282_ref044",
      "first-page": "164",
      "article-title": "A symplectic difference scheme for the infinite dimensional Hamiltonian system",
      "volume": "6",
      "author": "Chun-wang",
      "year": "1988",
      "journal-title": "J. Comput. Math."
    },
    {
      "key": "S0962492900002282_ref008",
      "doi-asserted-by": "publisher",
      "DOI": "10.1137/0716004"
    },
    {
      "key": "S0962492900002282_ref047",
      "volume-title": "Program in Applied Mathematics",
      "author": "McLachlan",
      "year": "1991"
    },
    {
      "key": "S0962492900002282_ref027",
      "first-page": "1",
      "volume-title": "Proceedings of the 1st Chinese Conference for Numerical Methods for PDE's, March 1986, Shanghai",
      "author": "Kang",
      "year": "1987"
    },
    {
      "key": "S0962492900002282_ref058",
      "doi-asserted-by": "publisher",
      "DOI": "10.1109/TNS.1983.4332919"
    },
    {
      "key": "S0962492900002282_ref072",
      "doi-asserted-by": "crossref",
      "first-page": "1041",
      "DOI": "10.1016/0898-1221(88)90139-3",
      "article-title": "The generating function for the solution of ODE's and its discrete methods",
      "volume": "15",
      "author": "Yuhua",
      "year": "1988",
      "journal-title": "Comput. Math. Appl."
    },
    {
      "key": "S0962492900002282_ref061",
      "volume-title": "Applied Mathematics and Computation Reports",
      "author": "Sanz-Serna",
      "year": "1990"
    },
    {
      "key": "S0962492900002282_ref049",
      "volume-title": "Schwerpunktprogramm der Deutschen Forschungsgemeinschaft, Anwendungsbezogene Optimierung und Steuerung",
      "author": "Miesbach",
      "year": "1990"
    },
    {
      "key": "S0962492900002282_ref006",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/978-3-662-06793-2"
    },
    {
      "key": "S0962492900002282_ref055",
      "doi-asserted-by": "publisher",
      "DOI": "10.1098/rspa.1991.0027"
    },
    {
      "key": "S0962492900002282_ref015",
      "doi-asserted-by": "publisher",
      "DOI": "10.1016/0021-9991(91)90299-Z"
    },
    {
      "key": "S0962492900002282_ref071",
      "doi-asserted-by": "publisher",
      "DOI": "10.1016/0375-9601(90)90092-3"
    },
    {
      "key": "S0962492900002282_ref035",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/978-1-4612-1140-2"
    },
    {
      "key": "S0962492900002282_ref043",
      "unstructured": "Lasagni F. (1990), \u2018Integration methods for Hamiltonian differential equations\u2019, unpublished manuscript."
    },
    {
      "key": "S0962492900002282_ref067",
      "doi-asserted-by": "publisher",
      "DOI": "10.1137/0147005"
    },
    {
      "key": "S0962492900002282_ref029",
      "first-page": "71",
      "article-title": "Construction of canonical difference schemes for Hamiltonian formalism via generating functions",
      "volume": "7",
      "author": "Kang",
      "year": "1989",
      "journal-title": "J. Comput. Math."
    },
    {
      "key": "S0962492900002282_ref031",
      "volume-title": "Applied Mathematics and Computation Reports",
      "author": "de Frutos",
      "year": "1991"
    },
    {
      "key": "S0962492900002282_ref059",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/BF01954907"
    },
    {
      "key": "S0962492900002282_ref034",
      "doi-asserted-by": "publisher",
      "DOI": "10.1137/0907067"
    },
    {
      "key": "S0962492900002282_ref052",
      "unstructured": "Neri F. (1987), \u2018Lie algebras and canonical integration\u2019, University of Maryland Technical Report."
    },
    {
      "key": "S0962492900002282_ref038",
      "doi-asserted-by": "publisher",
      "DOI": "10.1093/imanum/10.1.1"
    },
    {
      "key": "S0962492900002282_ref039",
      "volume-title": "Numerical Analysis Report DAMTP",
      "author": "Iserles",
      "year": "1990"
    },
    {
      "key": "S0962492900002282_ref068",
      "first-page": "202",
      "article-title": "Canonical transformations generated by methods of Runge-Kutta type for the numerical integration of the system x\u2033 = \u2212\u2202U\u2202x",
      "volume": "29",
      "author": "Suris",
      "year": "1989",
      "journal-title": "Zh. Vychisl. Mat. i Mat. Fiz."
    },
    {
      "key": "S0962492900002282_ref033",
      "doi-asserted-by": "publisher",
      "DOI": "10.1016/0375-9601(88)90773-6"
    },
    {
      "key": "S0962492900002282_ref012",
      "article-title": "Order conditions for canonical Runge\u2013Kutta\u2013Nystr\u00f6m methods",
      "author": "Calvo",
      "year": "1991",
      "journal-title": "BIT"
    },
    {
      "key": "S0962492900002282_ref016",
      "unstructured": "Channell P. J. , \u2018Symplectic integration algorithms\u2019, Los Alamos National Laboratory Report, Report AT-6ATN 83\u20139."
    },
    {
      "key": "S0962492900002282_ref028",
      "first-page": "371",
      "article-title": "Symplectic difference schemes for linear Hamiltonian canonical systems",
      "volume": "8",
      "author": "Kang",
      "year": "1990",
      "journal-title": "J. Comput. Math."
    },
    {
      "key": "S0962492900002282_ref009",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/BF01932265"
    },
    {
      "key": "S0962492900002282_ref007",
      "doi-asserted-by": "crossref",
      "first-page": "175",
      "DOI": "10.1093/oso/9780198534389.003.0005",
      "volume-title": "Advances in Numerical Analysis",
      "volume": "I",
      "author": "Beyn",
      "year": "1991"
    },
    {
      "key": "S0962492900002282_ref004",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/978-3-662-02535-2"
    },
    {
      "key": "S0962492900002282_ref037",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/978-3-662-12607-3"
    },
    {
      "key": "S0962492900002282_ref021",
      "volume-title": "Stability of Runge\u2013Kutta Methods for Stiff Nonlinear Differential Equations",
      "author": "Dekker",
      "year": "1984"
    },
    {
      "key": "S0962492900002282_ref064",
      "doi-asserted-by": "publisher",
      "DOI": "10.1137/0728058"
    },
    {
      "key": "S0962492900002282_ref022",
      "doi-asserted-by": "publisher",
      "DOI": "10.1093/imanum/7.2.235"
    },
    {
      "key": "S0962492900002282_ref024",
      "first-page": "42",
      "volume-title": "Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations",
      "author": "Kang",
      "year": "1985"
    },
    {
      "key": "S0962492900002282_ref032",
      "doi-asserted-by": "publisher",
      "DOI": "10.1016/0045-7825(90)90046-O"
    },
    {
      "key": "S0962492900002282_ref020",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/BF01397651"
    },
    {
      "key": "S0962492900002282_ref011",
      "unstructured": "Calvo M.P. (1991), Ph.D. Thesis, University of Valladolid (to appear)."
    },
    {
      "key": "S0962492900002282_ref026",
      "first-page": "1",
      "volume-title": "Tenth International Conference on Numerical Methods in Fluid Dynamics",
      "author": "Kang",
      "year": "1986"
    },
    {
      "key": "S0962492900002282_ref001",
      "volume-title": "Applied Mathematics and Computation Reports",
      "author": "Abia",
      "year": "1990"
    },
    {
      "key": "S0962492900002282_ref002",
      "doi-asserted-by": "publisher",
      "DOI": "10.1016/0021-9991(85)90180-9"
    },
    {
      "key": "S0962492900002282_ref003",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/978-3-642-61551-1"
    },
    {
      "key": "S0962492900002282_ref005",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/978-1-4757-2063-1"
    },
    {
      "key": "S0962492900002282_ref054",
      "first-page": "91",
      "volume-title": "Working Document",
      "author": "Okunbor",
      "year": "1991"
    },
    {
      "key": "S0962492900002282_ref010",
      "volume-title": "The Numerical Analysis of Ordinary Differential Equations",
      "author": "Butcher",
      "year": "1987"
    },
    {
      "key": "S0962492900002282_ref014",
      "volume-title": "Applied Mathematics and Computation Reports",
      "author": "Calvo",
      "year": "1991"
    },
    {
      "key": "S0962492900002282_ref017",
      "doi-asserted-by": "publisher",
      "DOI": "10.1088/0951-7715/3/2/001"
    },
    {
      "key": "S0962492900002282_ref013",
      "volume-title": "Applied Mathematics and Computation Reports",
      "author": "Calvo",
      "year": "1991"
    },
    {
      "key": "S0962492900002282_ref018",
      "doi-asserted-by": "publisher",
      "DOI": "10.1093/imanum/7.1.1"
    },
    {
      "key": "S0962492900002282_ref019",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/BF02238800"
    },
    {
      "key": "S0962492900002282_ref023",
      "volume-title": "Applied Mathematics and Computation Reports",
      "author": "Eirola",
      "year": "1990"
    },
    {
      "key": "S0962492900002282_ref025",
      "first-page": "279",
      "article-title": "Difference schemes for Hamiltonian formalism and symplectic geometry",
      "volume": "4",
      "author": "Kang",
      "year": "1986",
      "journal-title": "J. Comput. Math."
    },
    {
      "key": "S0962492900002282_ref030",
      "doi-asserted-by": "publisher",
      "DOI": "10.1016/0167-2789(90)90019-L"
    },
    {
      "key": "S0962492900002282_ref036",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/BF01385623"
    },
    {
      "key": "S0962492900002282_ref040",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/978-1-4899-3071-2"
    },
    {
      "key": "S0962492900002282_ref041",
      "volume-title": "Vorlesungen \u00dcber die Entwicklung der Mathematik im 19.Jahrhundert",
      "author": "Klein",
      "year": "1926"
    },
    {
      "key": "S0962492900002282_ref042",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/BF00945133"
    }
  ],
  "container-title": "Acta Numerica",
  "original-title": [],
  "language": "en",
  "link": [
    {
      "URL": "https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0962492900002282",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "similarity-checking"
    }
  ],
  "deposited": {
    "date-parts": [
      [
        2024,
        3,
        2
      ]
    ],
    "date-time": "2024-03-02T09:52:12Z",
    "timestamp": 1709373132000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "https://www.cambridge.org/core/product/identifier/S0962492900002282/type/journal_article"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        1992,
        1
      ]
    ]
  },
  "references-count": 72,
  "alternative-id": [
    "S0962492900002282"
  ],
  "URL": "http://dx.doi.org/10.1017/S0962492900002282",
  "relation": {},
  "ISSN": [
    "0962-4929",
    "1474-0508"
  ],
  "subject": [],
  "container-title-short": "Acta Numerica",
  "published": {
    "date-parts": [
      [
        1992,
        1
      ]
    ]
  }
}