{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T20:25:16Z","timestamp":1782937516697,"version":"3.54.5"},"reference-count":163,"publisher":"Association for Computing Machinery (ACM)","issue":"1","license":[{"start":{"date-parts":[[1985,3,1]],"date-time":"1985-03-01T00:00:00Z","timestamp":478483200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Comput. Surv."],"published-print":{"date-parts":[[1985,3]]},"abstract":"<jats:p>Mathematical models when simulating the behavior of physical, chemical, and biological systems often include one or more ordinary differential equations (ODEs). To study the system behavior predicted by a model, these equations are usually solved numerically.<\/jats:p>\n          <jats:p>Although many of the current methods for solving ODEs were developed around the turn of the century, the past 15 years or so has been a period of intensive research. The emphasis of this survey is on the methods and techniques used in software for solving ODEs.<\/jats:p>\n          <jats:p>ODEs can be classified as stiff or nonstiff, and may be stiff for some parts of an interval and nonstiff for others. We discuss stiff equations, why they are difficult to solve, and methods and software for solving both nonstiff and stiff equations. We conclude this review by looking at techniques for dealing with special problems that may arise in some ODEs, for example, discontinuities.<\/jats:p>\n          <jats:p>Although important theoretical developments have also taken place, we report only those developments which have directly affected the software and provide a review of this research. We present the basic concepts involved but assume that the reader has some background in numerical computing, such as a first course in numerical methods.<\/jats:p>","DOI":"10.1145\/4078.4079","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T11:32:04Z","timestamp":1027769524000},"page":"5-47","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":49,"title":["A review of recent developments in solving ODEs"],"prefix":"10.1145","volume":"17","author":[{"given":"Gopal K.","family":"Gupta","sequence":"first","affiliation":[{"name":"Monash Univ., Clayton, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ron","family":"Sacks-Davis","sequence":"additional","affiliation":[{"name":"Melbourne Institute of Technology, Melbourne, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peter E.","family":"Tescher","sequence":"additional","affiliation":[{"name":"Monash Univ., Clayton, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[1985,3]]},"reference":[{"key":"e_1_2_1_2_1","volume-title":"Proceedings of the International Conference on Sti# Computation (Park City, Utah, Apr.). To be published in Stiff Computation","author":"AIKEN R. C.","year":"1985","unstructured":"AIKEN , R. C. 1982. Stiff review 1974-1982: 1. Applications . In Proceedings of the International Conference on Sti# Computation (Park City, Utah, Apr.). To be published in Stiff Computation , R. C. Aiken, Ed. Oxford Univ. Press , London and New York, 1985 .]] AIKEN, R. C. 1982. Stiff review 1974-1982: 1. Applications. In Proceedings of the International Conference on Sti# Computation (Park City, Utah, Apr.). To be published in Stiff Computation, R. C. Aiken, Ed. Oxford Univ. Press, London and New York, 1985.]]"},{"key":"e_1_2_1_3_1","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1007\/BF01432876","article-title":"On the order of composite multistep methods for ordinary differential equations","volume":"29","author":"ALBRECHT P.","year":"1978","unstructured":"ALBRECHT , P. 1978 . On the order of composite multistep methods for ordinary differential equations . Numer. Math. 29 , 381 - 396 .]] ALBRECHT, P. 1978. On the order of composite multistep methods for ordinary differential equations. Numer. Math. 29, 381-396.]]","journal-title":"Numer. Math."},{"key":"e_1_2_1_4_1","doi-asserted-by":"crossref","first-page":"1006","DOI":"10.1137\/0714068","article-title":"Diagonally implicit Runge-Kutta methods for stiff O.D.E.'s","volume":"14","author":"ALEXANDER R.","year":"1977","unstructured":"ALEXANDER , R. 1977 . Diagonally implicit Runge-Kutta methods for stiff O.D.E.'s . SINUM 14 , 1006 - 1021 .]] ALEXANDER, R. 1977. Diagonally implicit Runge-Kutta methods for stiff O.D.E.'s. SINUM 14, 1006-1021.]]","journal-title":"SINUM"},{"key":"e_1_2_1_5_1","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1137\/0716045","article-title":"Numerical solution of solution of ordinary differential equations separated into subsystems","volume":"16","author":"ANDRUS J. F.","year":"1979","unstructured":"ANDRUS , J. F. 1979 . Numerical solution of solution of ordinary differential equations separated into subsystems . SIAM J. Numer. Anal. 16 , 605 - 611 .]] ANDRUS, J. F. 1979. Numerical solution of solution of ordinary differential equations separated into subsystems. SIAM J. Numer. Anal. 16, 605-611.]]","journal-title":"SIAM J. Numer. Anal."},{"key":"e_1_2_1_6_1","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1007\/BF01418331","article-title":"A semi-implicit mid-point rule for stiff differential systems of ordinary differential equations","volume":"41","author":"BADER G.","year":"1983","unstructured":"BADER , G. , AND DEUFLHARD , P. 1983 . A semi-implicit mid-point rule for stiff differential systems of ordinary differential equations . Numer. Math. 41 , 373 - 398 .]] BADER, G., AND DEUFLHARD, P. 1983. A semi-implicit mid-point rule for stiff differential systems of ordinary differential equations. Numer. Math. 41,373-398.]]","journal-title":"Numer. 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H. , AND HULL , T. E. 1975 . STIFF DETEST: A program for comparing numerical methods for stiff ordinary differential equations. Tech. Rep. 81, Dept. of Computer Science, Univ. of Toronto, Ont., Canada.]] BEDET, R. A., ENRIGHT, W. H., AND HULL, T. E. 1975. STIFF DETEST: A program for comparing numerical methods for stiff ordinary differential equations. Tech. Rep. 81, Dept. of Computer Science, Univ. of Toronto, Ont., Canada.]]"},{"key":"e_1_2_1_10_1","series-title":"Lecture Notes in Mathematics","first-page":"9","volume-title":"Numerical Treatment of Differential Equations","author":"BETTIS D. G.","unstructured":"BETTIS , D. G. 1978. Efficient embedded Runge-Kutta methods . In Numerical Treatment of Differential Equations , Lecture Notes in Mathematics , vol. 631 . Springer-Verlag , Berlin and New York, pp. 9 - 18 .]] BETTIS, D. G. 1978. Efficient embedded Runge-Kutta methods. In Numerical Treatment of Differential Equations, Lecture Notes in Mathematics, vol. 631. 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N. , AND HINDMARSH , A. C. 1984. Matrix free methods for stiff ODEs. Rep. UCRL- 90770 , Lawrence Livermore National Laboratory , Lawrence, Calif ., May 1984 .]] 10.1137\/0723039 BROWN, P. N., AND HINDMARSH, A. C. 1984. Matrix free methods for stiff ODEs. Rep. UCRL- 90770, Lawrence Livermore National Laboratory, Lawrence, Calif., May 1984.]] 10.1137\/0723039"},{"key":"e_1_2_1_17_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02165234","article-title":"Numerical treatment of ordinary differential equations by extrapolation methods","volume":"8","author":"BULIRSCH","year":"1966","unstructured":"BULIRSCH , g., AND STOER , J. 1966 . Numerical treatment of ordinary differential equations by extrapolation methods . Numer. Math. 8 , 1 - 13 .]] BULIRSCH, g., AND STOER, J. 1966. Numerical treatment of ordinary differential equations by extrapolation methods. Numer. Math. 8, 1-13.]]","journal-title":"Numer. 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An implementation of singly-implicit Runge-Kutta methods. BIT 20, 326-340.]]","journal-title":"BIT"},{"key":"e_1_2_1_20_1","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1090\/S0025-5718-1964-0159424-9","article-title":"Implicit Runge-Kutta processes","volume":"18","author":"BUTCHER J. C.","year":"1964","unstructured":"BUTCHER , J. C. 1964 a. Implicit Runge-Kutta processes . Math. Comput. 18 , 50 - 64 .]] BUTCHER, J. C. 1964a. Implicit Runge-Kutta processes. Math. Comput. 18, 50-64.]]","journal-title":"Math. Comput."},{"key":"e_1_2_1_21_1","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1090\/S0025-5718-1964-0165693-1","article-title":"integration processes based on Radau quadrature formulas","volume":"18","author":"BUTCHER J. C.","year":"1964","unstructured":"BUTCHER , J. C. 1964 b. integration processes based on Radau quadrature formulas . Math. Comput. 18 , 233 - 243 .]] BUTCHER, J. C. 1964b. integration processes based on Radau quadrature formulas. Math. 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Academic Press, Orlando, Fla., pp. 243-265.]]"},{"key":"e_1_2_1_27_1","first-page":"153","volume-title":"Proceedings o{ the International Con{erence SIMULATION 77","author":"CARVER M. B.","year":"1977","unstructured":"CARVER , M. B. 1977 . Efficient handling of discontinuities and time delays in ordinary differential equation systems . In Proceedings o{ the International Con{erence SIMULATION 77 ( Montreaux, Switzerland) , pp. 153 - 158 .]] CARVER, M. B. 1977. Efficient handling of discontinuities and time delays in ordinary differential equation systems. In Proceedings o{ the International Con{erence SIMULATION 77 (Montreaux, Switzerland), pp. 153-158.]]"},{"key":"e_1_2_1_28_1","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1016\/0378-4754(78)90068-X","article-title":"Efficient implementation over discontinuities in ordinary differential equation simulations","volume":"20","author":"CARVER M. B.","year":"1978","unstructured":"CARVER , M. B. 1978 . Efficient implementation over discontinuities in ordinary differential equation simulations . Math. Comput. Simulat. 20 , 190 - 196 .]] CARVER, M. B. 1978. Efficient implementation over discontinuities in ordinary differential equation simulations. Math. Comput. Simulat. 20, 190-196.]]","journal-title":"Math. Comput. Simulat."},{"key":"e_1_2_1_29_1","volume-title":"Rep. AECL-5177","author":"CARVER M. B.","year":"1976","unstructured":"CARVER , M. B. , AND BAUDOUIN , A. P. 1976. Solution of reactor kinetics problems using sparse matrix techniques in an ODE integrator for stiff equations. Atomic Energy of Canada Limited , Rep. AECL-5177 , Jan. 1976 .]] CARVER, M. B., AND BAUDOUIN, A. P. 1976. Solution of reactor kinetics problems using sparse matrix techniques in an ODE integrator for stiff equations. Atomic Energy of Canada Limited, Rep. 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BIT 11,384-388.]]","journal-title":"BIT"},{"key":"e_1_2_1_38_1","doi-asserted-by":"publisher","DOI":"10.1145\/1271.1610"},{"key":"e_1_2_1_39_1","first-page":"541","article-title":"Semi-explicit, A-stable Runge-Kutta methods","volume":"33","author":"COOPER J. F.","year":"1979","unstructured":"COOPER , J. F. , AND SAYFY , A. 1979 . Semi-explicit, A-stable Runge-Kutta methods . Math. Comput. 33 , 541 - 556 .]] COOPER, J. F., AND SAYFY, A. 1979. Semi-explicit, A-stable Runge-Kutta methods. Math. Comput. 33, 541-556.]]","journal-title":"Math. Comput."},{"key":"e_1_2_1_40_1","doi-asserted-by":"publisher","DOI":"10.1145\/355993.355995"},{"key":"e_1_2_1_41_1","volume-title":"Computational Techniques for Ordinary Differential Equations","author":"CURTIS A. R.","unstructured":"CURTIS , A. R. 1980. The FACSIMILE numerical integrator for stiff ordinary differential problems . In Computational Techniques for Ordinary Differential Equations , I. Gladwell and D. K. Sayers, Eds. Academic Press , Orlando, Fla .]] CURTIS, A. R. 1980. The FACSIMILE numerical integrator for stiff ordinary differential problems. In Computational Techniques for Ordinary Differential Equations, I. Gladwell and D. K. Sayers, Eds. Academic Press, Orlando, Fla.]]"},{"key":"e_1_2_1_42_1","first-page":"117","article-title":"On the estimation of sparse Jacobian matrices","volume":"13","author":"CURTIS A. R.","year":"1974","unstructured":"CURTIS , A. R. , POWELL , M. J. D., AND REID , J. X. 1974 . On the estimation of sparse Jacobian matrices . J. IMA 13 , 117 - 119 .]] CURTIS, A. R., POWELL, M. J. D., AND REID, J. X. 1974. On the estimation of sparse Jacobian matrices. J. IMA 13, 117-119.]]","journal-title":"J. IMA"},{"key":"e_1_2_1_43_1","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1073\/pnas.38.3.235","article-title":"Integration of stiff equations","author":"CURTIS C. E.","year":"1952","unstructured":"CURTIS , C. E. , AND HIRSCHFELDER , J. O. 1952 . 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