Runge-Kutta methods: Some historical notes

Applied Numerical Mathematics

Volumn 22, Issue 1-3 SPEC. ISS., 1996, Pages 113-151

  Butcher, J C ^a   Wanner, G ^b  

^a UNIVERSITY OF AUCKLAND   (New Zealand)

^b UNIVERSITY OF GENEVA   (Switzerland)

Author keywords

Composition; Hamiltonian systems; History; Order conditions; Order reduction; Runge kutta methods; Stability

Indexed keywords

DIFFERENTIAL EQUATIONS; NONLINEAR EQUATIONS;

HAMILTONIAN SYSTEMS; RUNGE KUTTA METHODS;

NUMERICAL METHODS;

EID: 0030285953     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/s0168-9274(96)00048-7     Document Type: Article

References (129)

1. J.C. Adams. Appendix in: F. Bashforth, An Attempt to Test the Theories of Capillary Action by Comparing the Theoretical and Measured Forms of Drops of Fluid. With an Explanation of the Method of Integration Employed in Constructing the Tables Which Give the Theoretical Form of Such Drops, by J.C. Adams (Cambridge University Press, Cambridge, 1883).
2. P. Albrecht, Numerical treatment of O.D.Es.: The theory of A-methods, Numer. Math. 17 (1985) 59-87.
3. R. Alexander, Diagonally implicit Runge-Kutta methods for stiff O.D.E.'s. SIAM J. Numer. Anal. 14 (1977) 1006-1021.
4. R. Alt, Méthodes A-stables pour l'intégration de systèmes différentiels mal conditionnés, Thèse, Univ. Paris VI (1971).
5. O. Axelsson, A class of A-stable methods, BIT 9 (1969) 185-199.
6. G. Bader and P. Deuflhard, A semi-implicit mid-point rule for stiff systems of ordinary differential equations, Numer. Math. 41 (1983) 373-398.
7. C.T.H. Baker, The state of the art in the numerical treatment of integral equations, in: A. Iserles and M.J.D. Powell, eds., The State of the Art in Numerical Analysis (Clarendon Press, Oxford, 1987) 473-509.
8. G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms, J. Statist. Phys. 74 (1994) 1117-1143.
9. L. Bieberbach, On the remainder of the Runge-Kutta formula in the theory of ordinary differential equations, Z. Angew. Math. Phys. 2 (1951) 233-248.
10. G. Birkhoff and R.S. Varga, Discretization errors for well-set Cauchy problems I, J. Math. Phys. 44 (1965) 1-23.
11. H. Brunner and P.J. van der Houwen, The Numerical Solution of Volterra Equations, CWI Monographs 3 (North-Holland, Amsterdam, 1986).
12. R. Bulirsch and M. Breitner, Wilhelm Martin Kutta (1867-1944), DMV Mitteilungen 2 (1994) 7-8.
13. R. Bulirsch and J. Stoer, Numerical treatment of ordinary differential equations by extrapolation methods, Numer. Math. 8 (1966) 1-13.
14. K. Burrage, J.C. Butcher and F.H. Chipman, An implementation of singly-implicit Runge-Kutta methods, BIT 20 (1980) 326-340.
15. K. Burrage and J.C. Butcher, Stability criteria for implicit Runge-Kutta methods, SIAM J. Numer. Anal. 16 (1979) 46-57.
16. J.C. Butcher, Coefficients for the study of Runge-Kutta integration processes, J. Austral. Math. Soc. 3 (1963) 185-201.
17. J.C. Butcher, On Runge-Kutta methods of high order. J. Austral. Math. Soc. 4 (1964) 179-194.
18. J.C. Butcher, Implicit Runge-Kutta processes. Math. Comp. 18 (1964) 50-64.
19. J.C. Butcher, Integration processes based on Radau quadrature formulas, Math. Comp. 18 (1964) 233-244.
20. J.C. Butcher, On the attainable order of Runge-Kutta methods, Math. Comp. 19 (1965) 408-417.
21. J.C. Butcher, An algebraic theory of integration methods, Math. Comp. 26 (1972) 79-106.
22. J.C. Butcher, A stability property of implicit Runge-Kutta methods, BIT 15 (1975) 358-361.
23. J.C. Butcher, The non-existence of ten stage eighth order explicit Runge-Kutta methods, BIT 25 (1985) 521-540.
24. J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods (Wiley, Chichester, 1987).
25. J.C. Butcher, A history of Runge-Kutta methods, Appl. Numer. Math. 20 (1996) 247-260.
26. A.L. Cauchy, Résumé des Leçons données à l'Ecole Royale Polytechnique. Suite du Calcul Infinitésimal (1824); published: Chr. Gilain and Johnson, eds, Equations Différentielles Ordinaires (1981).
27. A.L. Cauchy, Résumé des Leçons données à l'Ecole Royale Polytechnique. Suite du Calcul Infinitésimal (1824); published: Chr. Gilain and Johnson, eds, Equations Différentielles Ordinaires (1981).
28. F.H. Chipman, A-stable Runge-Kutta processes, BIT 11 (1971) 384-388.
29. L. Collatz Numerische Behandlung von Differentialgleichungen, Grundlehren der Mathematischen Wissenschaften 60 (Springer, New York, 1951); 2nd ed. 1955; 3rd ed. and English translation 1960.
30. G.J. Cooper and J.H. Verner, Some explicit Runge-Kutta methods of high order, SIAM J. Numer. Anal. 9 (1972) 389-405.
31. G. Coriolis, Mémoire sur le degré d'approximation qu'on obtient pour les valeurs numériques d'une variable qui satisfait à une équation différentielle, en employant pour calculer ces valeurs diverses équations aux différences plus ou moins approchées, J. Math. Pures Appl. (Liouville) 2 (1837) 229-244.
32. R. Courant, K. Friedrichs and H. Lewy. Ueher die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann. 100 (1928) 32-74.
33. J. Crank and P. Nicolson, A practical method for numerical integration of solutions of partial differential equations of heat-conduction type, Proc. Cambridge Philos. Soc. 43 (1963) 50-67.
34. M. Crouzeix, Sur l'approximation des equations différentielles opérationnelles linéaires par de méthodes de Runge-Kutta. Thèse. Univ. Paris VI (1975).
35. M. Crouzeix, Sur la B-stabilité des méthodes de Runge-Kutta, Numer. Math. 32 (1979) 75-82.
36. M. Crouzcix and P.A. Raviart, Approximation des problèmes d'évolution, Unpublished lecture notes, Université de Rennes (1980).
37. M. Crouzcix, W.H. Hundsdorfer and M.N. Spijker. On the existence of solutions to the algebraic equations in implicit Runge-Kutta methods, BIT (1983) 84-91.
38. A.R. Curtis, An eighth order Runge-Kutta process with eleven function evaluations per step. Numer. Math. 16 (1970) 268-277.
39. A.R. Curtis, High order explicit Runge-Kutta formulae, their uses, and limitations, J. Inst. Math. Appl. 16 (1975) 35-55.
40. C.F. Curtiss and J.O. Hirschfelder. Integration of stiff equations, Proc. Nat. Acad. Sci. U.S.A. 38 (1952) 235-243.
41. G. Dahlquist, Fehlerabschätzungen bei Differenzenmethoden zur numerischen Integration gewöhnlicher Differentialgleichungen. Z. Angew. Math. Mech. 31 (1951) 239-240.
42. G. Dahlquist, A special stability problem for linear multistep methods, BIT 3 (1963) 27-43.
43. G. Dahlquist, Error analysis for a class of methods for stiff nonlinear initial value problems, in: Numerical Analysis, Dundee '75. Lecture Notes in Mathematics 506 (1975) 60-74.
44. G. Dahlquist, G-stability is equivalent to A-stability. BIT 18 (1978) 384-401.
45. G. Dahlquist, 33 years of numerical instability, part I. BIT 25 (1985) 188-204.
46. K. Dekker, On the iteration error in algebraically stable Runge-Kutta methods, Report NW 138/82, Math. Centrum, Amsterdam (1982).
47. K. Dekker, Error bounds for the solution to the algebraic equations in Runge-Kutta methods, BIT 24 (1984) 347-356.
48. K. Dekker, J.F.B.M. Kraaijevanger and J. Schneid, On the relation between algebraic stability and B-convergence for Runge-Kutta methods, Numer. Math. 51 (1990) 249-262.
49. K. Dekker and J.G. Verwer, Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations (North-Holland, Amsterdam, 1984).
50. J.R. Dormand and P.J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math. 6 (1980) 19-26.
51. B.L. Ehle, High order A-stable methods for the numerical solution of systems of DEs, BIT 8 (1968) 276-278.
52. B.L. Ehle, On Padé approximations to the exponential function and A-stable methods for the numerical solution of initial value problems, Research Report CSRR 2010, Dept. AACS, Univ. of Waterloo, Ontario, Canada (1969).
53. R. Emden, Gaskugeln, Anwendung der Mechanischen Wärmetheorie auf Kosmologische und Meteorologische Probleme (Teubner, Leipzig, 1907).
54. R. England, Error estimates for Runge-Kutta type solutions to systems of ordinary differential equations, Comput. J. 12 (1969) 166-170.
55. L. Euler, Institutiones Calculi Integralis. Volumen Primum (1768), Opera Omnia, Vol. XI, B.G. Teubneri, Lipsiae et Berolini MCMXIII.
56. E. Fehlberg, Classical fifth-, sixth-, seventh-, and eighth-order Runge-Kutta formulas with step size control, NASA Technical Report 287 (1968); extract published in Computing 4 (1969) 93-106.
57. E. Fehlberg, Classical fifth-, sixth-, seventh-, and eighth-order Runge-Kutta formulas with step size control, NASA Technical Report 287 (1968); extract published in Computing 4 (1969) 93-106.
58. Feng Kang, On difference schemes and symplectic geometry, Proceedings 5th Internat. Sympos. Differential Geom. Differential Equations, Beijing (August 1984) 42-58.
59. Feng Kang, Formal dynamical systems and numerical algorithms, talk at Internat. Conf. Comput. Differential Equations Dynamic Syst., Beijing (September 1992).
60. R. Frank, J. Schneid and C.W. Ueberhuber, The concept of B-convergence, SIAM J. Numer. Anal. 18 (1981) 753-780.
61. R. Frank, J. Schneid and C.W. Ueberhuber, Order results for implicit Runge-Kutta methods applied to stiff systems, SIAM J. Numer. Anal. 22 (1985) 515-534.
62. C.W. Gear, Runge-Kutta starters for multistep methods, ACM Trans. Math. Software 6 (1980) 263-279.
63. S. Gill, A process for the step-by-step integration of differential equations in an automatic digital computing machine, Proc. Cambridge Philos. Soc. 47 (1951) 95-108.
64. A. Guillou and B. Lago, Domaine de stabilité associé aux formules d'intégration numérique d'équations différentielles, à pas séparés et àpas liés. Recherche de formules à grand rayon de stabilité, ler Congr. Assoc. Fran. Calcul, AFCAL, Grenoble (September 1960) 43-56.
65. A. Guillou and J.L. Soulé, La résolution numérique des problèmes différentiels aux conditions initiales par des méthodes de collocation, R.I.R.O. No. R-3 (1969) 17-44.
66. E. Hairer, A Runge-Kutta method of order 10, J. Inst. Math. Appl. 21 (1978) 47-59.
67. E. Hairer, A- and B-stability for Runge-Kutta methods - characterizations and equivalence, Numer. Math. 48 (1986) 383-389.
68. E. Hairer, Backward analysis of numerical integrators and symplectic methods, Ann. Numer. Math. 1 (1994) 107-132.
69. E. Hairer and Ch. Lubich, The life-span of backward error analysis for numerical integrators, Numer. Math., to appear.
70. E. Hairer, Ch. Lubich and M. Roche, The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, Lecture Notes in Mathematics 1409 (Springer, Berlin, 1989).
71. E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, Springer Series in Computational Mathematics 8 (Springer, Berlin, 2nd ed., 1993).
72. E. Hairer and H. Türke, The equivalence of B-stability and A-stability, BIT 24 (1984) 520-528.
73. E. Hairer and G. Wanner. Algebraically stable and implementable Runge-Kutta methods of high order, SIAM J. Numer. Anal. 18 (1981) 1098-1108.
74. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics 14 (Springer, Berlin, 2nd ed., 1996).
75. E. Hairer and G. Wanner, On the Butcher group and general multi-value methods, Computing 13 (1974) 1-15.
76. P.C. Hammer and J.W. Hollingsworth, Trapezoidal methods of approximating solutions of differential equations, MTAC 9 (1955) 92-96.
77. P. Henrici, Discrete Variable Methods in Ordinary Differential Equations (Wiley, New York, 1962).
78. K. Heun, Neue Methode zur approximativen Integration der Differentialgleichungen einer unabhängigen Veränderlichen, Z. Math. Phys. 45 (1900) 23-38.
79. M.K. Horn, Fourth and fifth-order scaled Runge-Kutta algorithms for treating dense output, SIAM J. Numer. Anal. 20 (1983) 558-568.
80. W.H. Hundsdorfer and M.N. Spijker, A note on B-stability of Runge-Kutta methods, Numer. Math. 36 (1981) 319-331.
81. A. Huťa, Une amélioration de la méthode de Runge-Kutta-Nyström pour la résolution numérique des équations différentielles du premier ordre, Acta Math. Univ. Comenian. 1 (1956) 201-224.
82. L. Jay, Symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems, SIAM J. Numer. Anal. 33 (1996) 368-387.
83. R. Jeltsch and O. Nevanlinna, Largest disk of stability of explicit Runge-Kutta methods, BIT 18 (1978) 500-502.
84. R. Jeltsch and O. Nevanlinna, Stability of explicit time discretizations for solving initial value problems, Numer. Math. 37 (1981) 61-91; corrigendum: Numer. Math. 39, p. 155.
85. R. Jeltsch and O. Nevanlinna, Stability of explicit time discretizations for solving initial value problems, Numer. Math. 37 (1981) 61-91; corrigendum: Numer. Math. 39, p. 155.
86. R. Jeltsch and O. Nevanlinna, Stability and accuracy of time discretizations for initial value problems, Numer. Math. 40 (1982) 245-296.
87. P.E. Kloeden and E. Platen, Numeric Solution of Stochastic Differential Equations, Applications of Mathematics 23 (Springer, Berlin, 1992; corr. 2nd printing, 1995).
88. J.F.B.M. Kraaijevanger, B-convergence of the implicit midpoint rule and the trapezoidal rule, BIT 25 (1985) 652-666.
89. J. Kuntzmann, Neuere Entwickelungen der Methode von Runge-Kutta, Z. Angew. Math. Mech. 41 (1961) 28-31.
90. M.A. Kurdi, Stable high order methods for time discretization of stiff differential equations, Thesis, University of California (1974).
91. W. Kutta. Beitrag zur näherungsweisen Integration totaler Differentialgleichungen, Zeitschr. Math. Phys. 46 (1901) 435-453.
92. P. Laasonen, Über eine Methode zur Lösung der Wärmeleitungsgleichung, Acta Math. 81, 309-317.
93. F.M. Lasagni, Canonical Runge-Kutta methods, Z. Angew. Math. Phys. 39 (1988) 952-953.
94. W. Liniger, Zur Stabilität der numerischen Integrationsmethoden für Differentialgleichungen, Thèse, Université de Lausanne (1956).
95. W.S. Loud, On the long-run error in the numerical solution of certain differential equations, J. Math. Phys. 28(1) (1949) 45-49.
96. C. Lubich and A. Ostermann, Runge-Kutta methods for parabolic equations and convolution quadrature, Math. Comp. 60 (1993) 105-131.
97. R.H. Merson, An operational method for the study of integration processes, Proceedings Sympos. Data Processing, Weapons Research Establishment, Salisbury, Australia (1957) 110-1-110-25.
98. F.R. Moulton, New Methods in Exterior Ballistics (University Chicago Press, Chicago, 1926).
99. S.P. Nørsett, Semi-explicit Runge-Kutta methods. Report No. 6/74, Dept. of Math., Univ. of Trondheim, Norway (1974).
100. E.J. Nyström, Ueber die numerische Integration von Differentialgleichungen, Acta Soc. Sci. Fenn. 50 (1925) 1-54.
101. H. Poincaré. Les Méthodes Nouvelles de la Mécanique Céleste III (Gauthier-Villars, Paris, 1899).
102. A. Prothero and A. Robinson, On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math. Comp. 28 (1974) 145-162.
103. A. Ralston, Runge-Kutta methods with minimum error bounds, Math. Comp. 16 (1962) 431-437; corr. 17. p. 488.
104. H.H. Robertson, The solution of a set of reaction rate equations, in: J. Walsh, ed., Numerical Analysis, an Introduction (Academic Press, New York, 1966) 178-182.
105. H.H. Rosenbrock, Some general implicit processes for the numerical solution of differential equations, Computer J. 5 (1962/63) 329-330.
106. C. Runge, Ueber die numerische Auflösung von Differentialgleichungen, Math. Ann. 46 (1895) 167-178.
107. C. Runge, Ueber die numerische Auflösung totaler Differentialgleichungen, Göttinger Nachr. (1905) 252-257.
108. C. Runge, Über eine Methode die partielle Differentialgleichung Δu = Constans numerisch zu integrieren, Z. Math. Phys. 56 (1908) 225-232.
109. C. Runge and H. König, Vorlesungen über Numerisches Rechnen, Grundlehren der Mathematischen Wissenschaften 11 (Springer Berlin, 1924).
110. R.D. Ruth, A canonical integration technique, IEEE Trans. Nuclear Sci. 30 (1983) 2669-2671.
111. H. Rutishauser, Über die Instabilität von Methoden zur Integration von gewöhnlichen Differentialgleichungen, Z. Angew. Math. Phys. 3 (1952) 65-74.
112. A. Sandu, J.G. Verwer, M. van Loon, G.R. Carmichael, F.A. Potra, D. Dabdud and J.H. Seinfeld, Benchmarking stiff ODE solvers for atmospheric chemistry problems I: implicit versus explicit, Report NM-R9603, CWI. Amsterdam (January 1966).
113. J.M. Sanz-Serna, Runge-Kutta schemes for Hamiltonian systems, BIT 28 (1988) 877-883.
114. J.M. Sanz-Serna, Symplectic integrators for Hamiltonian problems: an overview. Acta Numer. 1 (1992) 243-286.
115. J.M. Sanz-Serna, Geometric integration, in: Proc. State of the Art in Numer. Anal. (York, 1996), to appear.
116. J.M. Sanz-Serna and M.P. Calvo, Numerical Hamiltonian Problems, Applied Mathematics and Mathematical Computation 7 (Chapman & Hall, London, 1994).
117. L.F. Shampine, Numerical Solution of Ordinary Differential Equations (Chapman & Hall, London, 1994).
118. L.F. Shampine, H.A. Watts and S.M. Davenport. Solving nonstiff ordinary differential equations - The state of the art, SIAM Rev. 18 (1976) 376-410.
119. H.J. Stetter, Local estimation of the global discretization error, SIAM J. Numer. Anal. 8 (1971) 512-523.
120. A.M. Stuart and A.R. Humphries, Model problems in numerical stability theory for initial value problems, SIAM Rev. 36 (1994) 226-257.
121. Sun Geng, Symplectic partitioned Runge-Kutta methods, J. Comput. Math. 11 (1993) 365-372.
122. Y.B. Suris, The canonicity of mappings generated by Runge-Kutta type methods when integrating the systems ẍ = -∂U/∂x, Zh. Vychisl. Mat. i Mat. Fiz. 29 (1989) 202-211 (in Russian); English translation in: Comput. Math. Math. Phys. 29 (1991) 138-144.
123. Y.B. Suris, The canonicity of mappings generated by Runge-Kutta type methods when integrating the systems ẍ = -∂U/∂x, Zh. Vychisl. Mat. i Mat. Fiz. 29 (1989) 202-211 (in Russian); English translation in: Comput. Math. Math. Phys. 29 (1991) 138-144.
124. J.H. Verner, Explicit Runge-Kutta methods with estimates of the local truncation error, SIAM J. Numer. Anal. 15 (1978) 772-790.
125. G. Wanner, E. Hairer and S.P. Nørsett, Order stars and stability theorems, BIT 18 (1978) 475-489; with an appendix BIT 18, 503.
126. G. Wanner, E. Hairer and S.P. Nørsett, Order stars and stability theorems, BIT 18 (1978) 475-489; with an appendix BIT 18, 503.
127. F.A. Willers, Methoden der Praktischen Analysis (Göschens Lehrbücherei, Berlin, 1928); later editions 1949, 1957; English translation: (Dover New York, 1947).
128. K. Wright, Some relationships between implicit Runge-Kutta collocation and Lanczos τ methods, and their stability properties, BIT 10 (1970) 217-227.
129. H. Yoshida, Recent progress in the theory and application of symplectic integrators, Celestial Mech. Dynam. Astronom. 56 (1993) 27-43.