Derivation and Implementation of Two-Step Runge-Kutta Pairs

Zdzislaw Jackiewicz, J. H. Verner

Research output: Contribution to journalArticlepeer-review

26 Scopus citations


Explicit Runge-Kutta pairs are known to provide efficient solutions to initial value differential equations with inexpensive derivative evaluations. Two-step Runge-Kutta methods strive to improve the efficiency by utilizing approximations to the solution and its derivatives from the previous step. This article suggests a strategy for computing embedded pairs of such two-step methods using a smaller number of function evaluations than that required for traditional Runge-Kutta methods of the same order. This leads to the efficient and reliable estimation of local discretization error and a robust step control strategy. The change of stepsize is achieved by a suitable interpolation of stage values from the previous step and does not require any additional function evaluations. Two examples illustrate the features of these pairs.

Original languageEnglish (US)
Pages (from-to)227-248
Number of pages22
JournalJapan Journal of Industrial and Applied Mathematics
Issue number2
StatePublished - Jun 2002


  • Implementation aspects
  • Local error estimation
  • Order conditions
  • Two-step Runge-Kutta methods

ASJC Scopus subject areas

  • Engineering(all)
  • Applied Mathematics


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