A history of Runge-Kutta methods

Applied Numerical Mathematics

Volumn 20, Issue 3, 1996, Pages 247-260

  Butcher, J C ^a  

^a UNIVERSITY OF AUCKLAND   (New Zealand)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; CONVERGENCE OF NUMERICAL METHODS; DIFFERENTIAL EQUATIONS; ERRORS; ESTIMATION;

ERROR ESTIMATION; RIEMANN RULE; RUNGE KUTTA METHODS; STABILITY ANALYSIS;

INTEGRATION;

EID: 0030101001     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/0168-9274(95)00108-5     Document Type: Article

References (36)

1. P. Albrecht, The extension of the theory of A-methods to RK methods, in: K. Strehmel, ed., Numerical Treatment of Differential Equations, Proceedings 4th Seminar NUMDIFF 4, Teubner Texte zur Mathematik (Teubner, Leipzig, 1987) 9-18.
2. R. Alexander, Diagonally implicit Runge-Kutta methods for stiff ODEs, SIAM J. Numer. Anal. 14 (1977) 1006-1021.
3. G. Birkhoff and R.S. Varga, Discretization errors for well-set Cauchy problems, J. Math. Phys. 44 (1965) 1-23.
4. K. Burrage, A special family of Runge-Kutta methods for solving stiff differential equations, BIT 18 (1978) 22-41.
5. K. Burrage and J.C. Butcher, Stability criteria for implicit Runge-Kutta methods, SIAM J. Numer. Anal. 16 (1979) 46-57.
6. J.C. Butcher, Coefficients for the study of Runge-Kutta integration processes, J. Austral. Math. Soc. 3 (1963) 185-201.
7. J.C. Butcher, On Runge-Kutta methods of high order, J. Austral. Math. Soc. 4 (1964) 179-194.
8. J.C. Butcher, Implicit Runge-Kutta processes, Math. Comp. 18 (1964) 50-64.
9. J.C. Butcher, The Numerical Analysis of Ordinary Differential Differential Equations: Runge-Kutta and General Linear Methods (Wiley, Chichester).
10. G.J. Cooper and J.H. Verner, Some explicit Runge-Kutta methods of high order, SIAM J. Numer. Anal. 9 (1972) 389-405.
11. M. Crouzeix, Sur la B-stabilité des méthodes de Runge-Kutta, Numer. Math. 32 (1979) 75-82.
12. A.R. Curtis, An eighth order Runge-Kutta process with eleven function evaluations per step, Numer. Math. 16 (1970) 268-277.
13. A.R. Curtis, High order explicit Runge-Kutta formulae, their uses, and limitations, J. Inst. Math. Appl. 16 (1975) 35-55.
14. J.R. Dormand and P.J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math. 6 (1980) 19-26.
15. E. Fehlberg, Klassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweiten-Kontrolle, Computing 4 (1969) 93-106.
16. R. Frank, J. Schneid and C.W. Ueberhuber, The concept of B-convergence, SIAM J. Numer. Anal. 18 (1981) 753-780.
17. R. Frank, J. Schneid and C.W. Ueberhuber, B-convergence: a survey, Appl. Numer. Math. 5 (1989) 51-61.
18. C.W. Gear, Runge-Kutta starters for multistep methods, ACM Trans. Math. Software 6 (1980) 263-279.
19. 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) 96-108.
20. E. Hairer, A Runge-Kutta method of order 10, J. Inst. Math. Appl. 21 (1978) 47-59.
21. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (Springer, Berlin, 1991).
22. P.C. Hammer and J.W. Hollingsworth, Trapezoidal methods of approximating solutions of differential equations, Math. Comp. 9 (1955) 92-96.
23. K. Heun, Neue Methoden zur approximativen Integration der Differentialgleichungen einer unabhängigen Veränderlichen, Z. Math. Phys. 45 (1900) 23-38.
24. 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 (1965) 21-24.
25. J. Kuntzmann, Neure Entwicklungen der Methoden von Runge und Kutta, Z. Angew. Math. Mech. 41 (1961) T28-T31.
26. W. Kutta, Beitrag zur näherungsweisen Integration totaler Differentialgleichungen, Z. Math. Phys. 46 (1901) 435-453.
27. D.J. Lambert, Numerical Methods for Ordinary Differential Systems: The Initial Value Problem (Wiley, Chichester, 1991).
28. R.H. Merson, An operational method for the study of integration processes, in: Proceedings Symposium on Data Processing, Salisbury, Australia (1957).
29. S.P. Nørsett, Semi-explicit Runge-Kutta methods, Report No. 6/74, Dept. of Math., Univ. of Trondheim, Norway (1974).
30. E.J. Nyström, Über die numerische Integration von Differentialgleichungen, Acta Soc. Sci. Fennicæ 50 (13) (1925) 55.
31. 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.
32. C. Runge, Über die numerische Auflösing von Differentialgleichungen, Math. Ann. 46 (1895) 167-178.
33. J.M. Sanz-Serna, Symplectic integrators for Hamiltonian problems: an overview, Acta Numer. 1 (1992) 243-286.
34. J.M. Sanz-Serna and M.P. Calvo, Numerical Hamiltonian Problems (Chapman and Hall, London, 1994).
35. J.H. Verner, Explicit Runge-Kutta methods with estimates of the local truncation error, SIAM J. Numer. Anal. 15 (1978) 772-790.
36. G. Wanner, E. Hairer and S.P. Nørsett, Order stars and stability theorems, BIT 18 (1978) 475-489.