A Family of FifthOrder RungeKutta Pairs
Authors:
S. N. Papakostas and G. Papageorgiou
Journal:
Math. Comp. 65 (1996), 11651181
MSC (1991):
Primary 65L05
MathSciNet review:
1333323
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Abstract: The construction of a RungeKutta pair of order with the minimal number of stages requires the solution of a nonlinear system of order conditions in unknowns. We define a new family of pairs which includes pairs using function evaluations per integration step as well as pairs which additionally use the first function evaluation from the next step. This is achieved by making use of Kutta's simplifying assumption on the original system of the order conditions, i.e., that all the internal nodes of a method contributing to the estimation of the endpoint solution provide, at these nodes, costfree secondorder approximations to the true solution of any differential equation. In both cases the solution of the resulting system of nonlinear equations is completely classified and described in terms of five free parameters. Optimal RungeKutta pairs with respect to minimized truncation error coefficients, maximal phaselag order and various stability characteristics are presented. These pairs were selected under the assumption that they are used in Local Extrapolation Mode (the propagated solution of a problem is the one provided by the fifthorder formula of the pair). Numerical results obtained by testing the new pairs over a standard set of test problems suggest a significant improvement in efficiency when using a specific pair of the new family with minimized truncation error coefficients, instead of some other existing pairs.
 1.
J.
C. Butcher, On RungeKutta processes of high order, J.
Austral. Math. Soc. 4 (1964), 179–194. MR 0165692
(29 #2972)
 2.
J.
C. Butcher, The numerical analysis of ordinary differential
equations, A WileyInterscience Publication, John Wiley & Sons,
Ltd., Chichester, 1987. Runge\mhy Kutta and general linear methods. MR 878564
(88d:65002)
 3.
C.
R. Cassity, The complete solution of the fifth order RungeKutta
equations, SIAM J. Numer. Anal. 6 (1969),
432–436. MR 0255061
(40 #8268)
 4.
A.
R. Curtis, Highorder explicit RungeKutta formulae, their uses,
and limitations, J. Inst. Math. Appl. 16 (1975),
no. 1, 35–55. MR 0383750
(52 #4630)
 5.
J.
R. Dormand and P.
J. Prince, A family of embedded RungeKutta formulae, J.
Comput. Appl. Math. 6 (1980), no. 1, 19–26. MR 568599
(81g:65098), http://dx.doi.org/10.1016/0771050X(80)900133
 6.
W. H. Enright and J. D. Pryce, Two FORTRAN packages for assessing initial value methods, ACM Trans. Math. Software 13 (1987), 127.
 7.
E. Fehlberg, Classical fifth, sixth, seventh, and eighth order RungeKutta formulas with stepsize control, TR R287, NASA, 1968.
 8.
, Low order classical RungeKutta formulas with stepsize control and their application to some heattransfer problems, TR R315, NASA, 1969.
 9.
E.
Hairer, S.
P. Nørsett, and G.
Wanner, Solving ordinary differential equations. I, Springer
Series in Computational Mathematics, vol. 8, SpringerVerlag, Berlin,
1987. Nonstiff problems. MR 868663
(87m:65005)
 10.
P.
J. van der Houwen and B.
P. Sommeijer, Explicit RungeKutta (Nyström) methods with
reduced phase errors for computing oscillating solutions, SIAM J.
Numer. Anal. 24 (1987), no. 3, 595–617. MR 888752
(88e:65088), http://dx.doi.org/10.1137/0724041
 11.
T.
E. Hull, W.
H. Enright, B.
M. Fellen, and A.
E. Sedgwick, Comparing numerical methods for ordinary differential
equations, SIAM J. Numer. Anal. 9 (1972),
603–637; errata, ibid. 11 (1974), 681. MR 0351086
(50 #3577)
 12.
W. Kutta, Beitrag zur näherungsweisen Integration totaler Differentialgleichungen, Z. Math. Phys. (1901), 435453.
 13.
J.
Douglas Lawson, An order five RungeKutta process with extended
region of stability, SIAM J. Numer. Anal. 3 (1966),
593–597. MR 0216760
(35 #7589)
 14.
G. Papageorgiou, Ch. Tsitouras, and S. N. Papakostas, RungeKutta pairs for periodic initial value problems, Rep. NA 931, Nat. Tech. Univ. Athens, Dept. Math., 1993.
 15.
S. N. Papakostas, Ch. Tsitouras, and G. Papageorgiou, A general family of explicit RungeKutta pairs of orders , SIAM J. Numer. Anal. 33 (1996).
 16.
L.
F. Shampine, Local extrapolation in the solution of
ordinary differential equations, Math.
Comp. 27 (1973),
91–97. MR
0331803 (48 #10135), http://dx.doi.org/10.1090/S00255718197303318031
 17.
E.
Baylis Shanks, Solutions of differential equations by
evaluations of functions, Math. Comp. 20 (1966), 21–38. MR 0187406
(32 #4858), http://dx.doi.org/10.1090/S00255718196601874061
 1.
 J. C. Butcher, On RungeKutta processes of high order, J. Austral. Math. Soc. 4 (1964), 179194. MR 29:2972
 2.
 , The numerical analysis of ordinary differential equations, John Wiley and Sons, Chichester, 1987. MR 88d:65002
 3.
 C. R. Cassity, The complete solution of the fifth order RungeKutta equations, SIAM J. Numer. Anal. 6 (1969), 432436. MR 40:8268
 4.
 A. R. Curtis, High order explicit RungeKutta formulae, their uses and their limitations, J. Inst. Math. Appl. 16 (1975), 3555. MR 52:4630
 5.
 J. R. Dormand and P. J. Prince, A family of embedded RungeKutta formulae, J. Comput. Appl. Math. 6 (1980), 1926. MR 81g:65098
 6.
 W. H. Enright and J. D. Pryce, Two FORTRAN packages for assessing initial value methods, ACM Trans. Math. Software 13 (1987), 127.
 7.
 E. Fehlberg, Classical fifth, sixth, seventh, and eighth order RungeKutta formulas with stepsize control, TR R287, NASA, 1968.
 8.
 , Low order classical RungeKutta formulas with stepsize control and their application to some heattransfer problems, TR R315, NASA, 1969.
 9.
 E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations I, first ed., Springer, Berlin, 1987. MR 87m:65005
 10.
 P. J. van der Houwen and B. P. Sommeijer, Explicit RungeKutta(Nyström) methods with reduced phase errors for computing oscillating solutions, SIAM J. Numer. Anal. 24 (1987), 595617. MR 88e:65088
 11.
 T. E. Hull, W. H. Enright, B. M. Fellen, and A. E. Sedgwick, Comparing numerical methods for ordinary differential equations, SIAM J. Numer. Anal. 9 (1972), 603637. MR 50:3577
 12.
 W. Kutta, Beitrag zur näherungsweisen Integration totaler Differentialgleichungen, Z. Math. Phys. (1901), 435453.
 13.
 J. D. Lawson, An order five RungeKutta process with extended region of stability, SIAM J. Numer. Anal. 3 (1966), 593597. MR 35:7589
 14.
 G. Papageorgiou, Ch. Tsitouras, and S. N. Papakostas, RungeKutta pairs for periodic initial value problems, Rep. NA 931, Nat. Tech. Univ. Athens, Dept. Math., 1993.
 15.
 S. N. Papakostas, Ch. Tsitouras, and G. Papageorgiou, A general family of explicit RungeKutta pairs of orders , SIAM J. Numer. Anal. 33 (1996).
 16.
 L. F. Shampine, Local extrapolation in the solution of ordinary differential equations, Math. Comp. 27 (1973), 9197. MR 48:10135
 17.
 B. E. Shanks, Solutions of differential equations by evaluations of functions, Math. Comp. 20 (1966), 2138. MR 32:4858
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Additional Information
S. N. Papakostas
Affiliation:
Department of Mathematics, Zografou Campus, National Technical University of Athens, Athens 157 80, Greece
Email:
spapakos@theseas.ntua.gr
G. Papageorgiou
Affiliation:
Department of Mathematics, Zografou Campus, National Technical University of Athens, Athens 157 80, Greece
Email:
papag@nisyros.ntua.gr
DOI:
http://dx.doi.org/10.1090/S0025571896007181
PII:
S 00255718(96)007181
Keywords:
Initial Value Problems,
RungeKutta,
pairs of embedded methods,
phaselag
Received by editor(s):
September 7, 1993
Received by editor(s) in revised form:
September 5, 1994, and April 5, 1995
Article copyright:
© Copyright 1996
American Mathematical Society
