Numerical comparisons of some explicit Runge-Kutta pairs of orders 4 through 8

ABSTRACT

We performed numerical testing of six explicit Runge-Kutta pairs ranging in order from a (3,4) pair to a (7,8) pair. All the test problems had smooth solutions and we assumed dense output was not required. The pairs were implemented in a uniform way. In particular, the stepsize selection for all pairs was based on the locally optimal formula. We tested the efficiency of the pairs, to what extent tolerance proportionality held, the accuracy of the local error estimate and stepsize prediction, and the performance on mildly stiff problems. We also showed, for these pairs, how the performance could be altered noticeably by making simple changes to the stepsize selection strategy. As part of the work, we demonstrated new ways of presenting numerical comparisons. —From the Author's Abstract

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INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.7 Ordinary Differential Equations
          Subjects: Initial value problems

Additional Classification:
  G. Mathematics of Computing
  G.4 MATHEMATICAL SOFTWARE
      Subjects: Certification and testing

General Terms:
Algorithms, Performance, Reliability

Keywords:
Runge-Kutta, efficiency explicit pairs, high order, low order