Interpolants for Runge-Kutta formulas

A general procedure for the construction of interpolants for Runge-Kutta (RK) formulas is presented. As illustrations, this approach is used to develop interpolants for three explicit RK formulas, including those employed in the well-known subroutines RKF45 and DVERK. A typical result is that no extra function evaluations are required to obtain an interpolant with <italic>O</italic>(<italic>h</italic><supscrpt>5</supscrpt>) local truncation error for the fifth-order RK formula used in RKF45; two extra function evaluations per step are required to obtain an interpolant with <italic>O</italic>(<italic>h</italic><supscrpt>6</supscrpt>) local truncation error for this RK formula.

[1]  Hans J. Stetter.  Interpolation and Error Estimation in Adams $PC$-Codes , 1979 .

[2]  Alfredo Bellen,et al.  Stability properties of interpolants for Runge-Kutta methods , 1988 .

[3]  T. E. Hull,et al.  Comparing Numerical Methods for Ordinary Differential Equations , 1972 .

[4]  L. S. Baca,et al.  Practical aspects of interpolation in Runge-Kutta codes , 1987 .

[5]  M. Zennaro.  Natural continuous extensions of Runge-Kutta formulas , 1986 .

[6]  M. K. Horn.  Scaled Runge-Kutta algorithms for handling dense output , 1981 .

[7]  C. William Gear.  Runge-Kutta Starters for Multistep Methods , 1980, TOMS.

[8]  Lawrence F. Shampine,et al.  Practical solution of ordinary differential equations by runge- kutta methods , 1976 .

[9]  H. A. Watts,et al.  DEPAC - design of a user oriented package of ODE solvers , 1980 .

[10]  M. Zennaro.  Natural continuous extensions of Runge-Kutta methods , 1986 .

[11]  J. Butcher.  Coefficients for the study of Runge-Kutta integration processes , 1963, Journal of the Australian Mathematical Society.

[12]  L. Shampine.  Interpolation for Runge–Kutta Methods , 1985 .

[13]  W. H. Enright,et al.  Effective solution of discontinuous IVPs using a Runge-Kutta formula pair with interpolants , 1988 .

[14]  P. J. Prince,et al.  Runge-Kutta triples , 1986 .

[15]  John D. Pryce,et al.  Two FORTRAN packages for assessing initial value methods , 1987, TOMS.

[16]  M. K. Horn,et al.  Fourth- and Fifth-Order, Scaled Rungs–Kutta Algorithms for Treating Dense Output , 1983 .

[17]  L. Shampine,et al.  Some practical Runge-Kutta formulas , 1986 .