Recently, pairs of explicit Runge–Kutta methods of orders 5 and 6 based on a new design have been derived independently by several authors. These pairs may be implemented so that the approximation of order 6 may be propagated using eight stages, and the extra derivative evaluation required for the error-estimating approximation of order 5 may be used again in the next step.An improved derivation of these pairs leads to a convenient generalization and thus to the derivation of families of higher-order pairs of the same design. For arbitrary p, pairs of orders $p - 1$ and p require $s \leq {{(p^2 - 7p + 22)} / 2}$ internal stages (the inequality holds for $p \geq 8$) and the derivative evaluation of the propagating stage. Furthermore, for each p, imposing an additional constraint on the nodes removes the necessity for the extra derivative evaluation. For each pair of this restricted subfamily, the method of order p utilizes all s stages so that the method of order $p - 1$ is properly imbedded. Examples of b...
[1]
J. H. Verner,et al.
Some Runge-Kutta formula pairs
,
1991
.
[2]
Péter Vértesi,et al.
Optimal lebesgue constant for lagrange interpolation
,
1990
.
[3]
J. Butcher.
The Numerical Analysis of Ordinary Di erential Equa-tions
,
1986
.
[4]
J. Verner.
Explicit Runge–Kutta Methods with Estimates of the Local Truncation Error
,
1978
.
[5]
John D. Pryce,et al.
Two FORTRAN packages for assessing initial value methods
,
1987,
TOMS.
[6]
J. Dormand,et al.
High order embedded Runge-Kutta formulae
,
1981
.
[7]
P. J. Prince,et al.
Global error estimation with runge-kutta triples
,
1989
.
[8]
Philip W. Sharp,et al.
Explicit Runge-Kutta Pairs with One More Derivative Evaluation than the Minimum
,
1993,
SIAM J. Sci. Comput..