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.

Cong N.H.

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Từ khóa : Approximation theory; Binary sequences; Computational methods; Differential equations; Error analysis; Parallel processing systems; Problem solving; Continuous approximation formulas; Continuous variable stepsize; Nonstiff problems; Sequential implementation mode; Runge Kutta methods