The aim of this paper is to apply a class of constant stepsize explicit pseudo two-step Runge-Kutta
methods of arbitrarily high order to nonstiff problems for systems of first-order differential equations with
variable stepsize strategy. Embedded formulas are provided for giving a cheap error estimate used in
stepsize control. Continuous approximation formulas are also considered for use in an eventual
implementation of the methods with dense output. By a few widely used test problems, we compare the
efficiency of two pseudo two-step Runge-Kutta methods of orders 5 and 8 with the codes DOPRI5, DOP853
and PIRK8. This comparison shows that in terms of f-evaluations on a parallel computer, these two pseudo
two-step Runge-Kutta methods are a factor ranging from 3 to 8 cheaper than DOPRI5, DOP853 and PIRK8.
Even in a sequential implementation mode, fifth-order new method beats DOPRI5 by a factor more than 1.5
with stringent error tolerances. ?? 1999 Elsevier Science B.V. All rights reserved.